10213:
31:
1601:
4036:
5411:
3760:
9811:
9850:(MDL) principle". "The description length of an input vector using a particular code is the sum of the code cost and reconstruction cost. They define this to be the energy of the code. Given an input vector, they define the energy of a code to be the sum of the code cost and the reconstruction cost." The true expected combined cost is
8125:
5216:
4031:{\displaystyle U\equiv \langle E\rangle =\sum _{r}P_{r}E_{r}=\sum _{r}{\frac {e^{-\beta E_{r}}E_{r}}{Z}}=\sum _{r}{\frac {-{\frac {\partial }{\partial \beta }}e^{-\beta E_{r}}}{Z}}={\frac {-{\frac {\partial }{\partial \beta }}\sum _{r}e^{-\beta E_{r}}}{Z}}=-{\frac {\partial \log Z}{\partial \beta }}.}
5608:
7651:
2498:
The laws of thermodynamics are only directly applicable to systems in thermal equilibrium. If we wish to describe phenomena like chemical reactions, then the best we can do is to consider suitably chosen initial and final states in which the system is in (metastable) thermal equilibrium. If the
7197:
4454:
9587:
9405:
6335:
7304:
3746:
of the system. The fact that the system does not have a unique energy means that the various thermodynamical quantities must be defined as expectation values. In the thermodynamical limit of infinite system size, the relative fluctuations in these averages will go to zero.
8544:
6876:
3003:
We see that the total amount of work that can be extracted in an isothermal process is limited by the free-energy decrease, and that increasing the free energy in a reversible process requires work to be done on the system. If no work is extracted from the system, then
1678:). The change in the Helmholtz energy during a process is equal to the maximum amount of work that the system can perform in a thermodynamic process in which temperature is held constant. At constant temperature, the Helmholtz free energy is minimized at equilibrium.
8814:
6598:
9159:
8402:
2629:
The volume of the system is kept constant. This means that the volume of the heat bath does not change either, and we can conclude that the heat bath does not perform any work. This implies that the amount of heat that flows into the heat bath is given by
8721:
8886:
7956:
4214:
6976:
2791:
5406:{\displaystyle S=\left.-\left({\frac {\partial F}{\partial T}}\right)\right|_{V,N},\quad P=\left.-\left({\frac {\partial F}{\partial V}}\right)\right|_{T,N},\quad \mu =\left.\left({\frac {\partial F}{\partial N}}\right)\right|_{T,V}.}
7945:
8308:
5476:
2878:
2289:
7540:
9821:
The
Helmholtz free energy function for a pure substance (together with its partial derivatives) can be used to determine all other thermodynamic properties for the substance. See, for example, the equations of state for
3438:
5727:
2957:
9806:{\displaystyle {\begin{aligned}F&={\frac {1}{2}}VC_{ijkl}\varepsilon _{ij}\varepsilon _{kl}-ST+\sum _{i}\mu _{i}N_{i}\\&={\frac {1}{2}}V\sigma _{ij}\varepsilon _{ij}-ST+\sum _{i}\mu _{i}N_{i}.\end{aligned}}}
7047:
4317:
2371:
2701:
7529:
9277:
8983:
2473:
6006:
4728:
2208:
2142:
6246:
7211:
4642:
4113:
7036:
3251:
8413:
6745:
6180:
9547:
1320:
3582:
2624:
9592:
6662:
8732:
1921:
4548:
7422:
6472:
8994:
4987:
4853:
2887:
is also the temperature of the system in these states. The fact that the system's temperature does not change allows us to express the numerator as the free energy change of the system:
8319:
2077:
7746:
Here we have used that the partition functions are constants with respect to taking averages and that the free energy is proportional to minus the logarithm of the partition function.
5206:
3734:
2036:
9913:
6464:
3626:
7804:
6042:
5800:
4910:
9229:
8618:
7741:
6376:
9468:
6209:
5906:
8610:
8164:
7343:
6733:
3314:. This equation is then again valid for both reversible and non-reversible changes. In case of a spontaneous change at constant T and V, the last term will thus be negative.
2998:
4790:
3037:
9265:
8120:{\displaystyle {\tilde {P}}_{r}=\left\langle r\left|{\frac {\exp \left}{\tilde {Z}}}\right|r\right\rangle ={\frac {\exp \left(-\beta {\tilde {E}}_{r}\right)}{\tilde {Z}}}\,}
9579:
6672:
For a classical model we can prove the
Bogoliubov inequality as follows. We denote the canonical probability distributions for the Hamiltonian and the trial Hamiltonian by
5466:
allow an efficient way of calculating thermodynamic variables of interest given the partition function and are often used in density of state calculations. One can also do
1784:
9438:
9191:
8253:
8224:
8193:
7853:
7776:
7703:
6408:
6238:
6102:
5852:
5759:
5672:
5461:
5087:
8825:
4937:
1155:
1100:
1045:
4484:
852:
805:
720:
673:
585:
538:
5857:
By including a large number of parameters in the trial
Hamiltonian and minimizing the free energy, we can expect to get a close approximation to the exact free energy.
3308:
2550:
2523:
1990:
1967:
756:
624:
4260:
5133:
4309:
4124:
2409:
990:
8574:
6887:
6697:
6435:
6129:
6073:
5936:
5014:
3654:
3495:
3468:
3281:
2716:
5040:
1697:
research
Helmholtz free energy is often used, since explosive reactions by their nature induce pressure changes. It is also frequently used to define fundamental
4283:
489:
7824:
7674:
5823:
5674:, which has different interactions and may depend on extra parameters that are not present in the original model. If we choose this trial Hamiltonian such that
5643:
4237:
3674:
2570:
1944:
828:
781:
696:
649:
561:
514:
10174:
7861:
5603:{\displaystyle m=\left.-\left({\frac {\partial F}{\partial B}}\right)\right|_{T,N},\quad V=\left.\left({\frac {\partial F}{\partial Q}}\right)\right|_{N,T}.}
8261:
7646:{\displaystyle \left\langle -\beta {\tilde {H}}-\log \left({\tilde {Z}}\right)\right\rangle \geq \left\langle -\beta H-\log \left(Z\right)\right\rangle }
2802:
2216:
1713:
1630:
7192:{\displaystyle \sum _{r}{\tilde {P}}_{r}\log \left({\frac {{\tilde {P}}_{r}}{P_{r}}}\right)\geq \sum _{r}\left({\tilde {P}}_{r}-P_{r}\right)=0\,}
3327:
5680:
4449:{\displaystyle d(\log Z)={\frac {\partial \log Z}{\partial \beta }}\,d\beta +{\frac {\partial \log Z}{\partial x}}\,dx=-U\,d\beta +\beta X\,dx.}
3145:, one needs to enlarge the thermodynamical state space of the system. In case of a chemical reaction, one must allow for changes in the numbers
5618:
Computing the free energy is an intractable problem for all but the simplest models in statistical physics. A powerful approximation method is
2893:
1810:
1331:
1219:
2300:
9400:{\displaystyle \mathrm {d} F=V\sum _{ij}\sigma _{ij}\,\mathrm {d} \varepsilon _{ij}-S\,\mathrm {d} T+\sum _{i}\mu _{i}\,\mathrm {d} N_{i},}
2636:
3524:
2502:
Since the thermodynamical variables of the system are well defined in the initial state and the final state, the internal energy increase
1708:, a German physicist, and first presented in 1882 in a lecture called "On the thermodynamics of chemical processes". From the German word
10229:
7433:
453:
10234:
10167:
8901:
6330:{\displaystyle \left\langle {\tilde {H}}\right\rangle ={\big \langle }H_{0}+\langle \Delta H\rangle {\big \rangle }=\langle H\rangle ,}
3743:
2417:
7299:{\displaystyle \left\langle \log \left({\tilde {P}}_{r}\right)\right\rangle \geq \left\langle \log \left(P_{r}\right)\right\rangle \,}
5944:
4653:
2150:
2089:
1309:
4559:
8539:{\displaystyle \sum _{r}{\tilde {P}}_{r}\log \left({\tilde {P}}_{r}\right)\geq \sum _{r}{\tilde {P}}_{r}\log \left(P_{r}\right)\,}
6871:{\displaystyle \sum _{r}{\tilde {P}}_{r}\log \left({\tilde {P}}_{r}\right)\geq \sum _{r}{\tilde {P}}_{r}\log \left(P_{r}\right)\,}
4055:
1342:
6987:
3164:
5143:
2499:
system is kept at fixed volume and is in contact with a heat bath at some constant temperature, then we can reason as follows.
6137:
10160:
9484:
912:
10111:
9831:
8809:{\displaystyle \left\langle \exp \left(X\right)\right\rangle _{r}\geq \exp \left(\left\langle X\right\rangle _{r}\right)\,}
3085:= constant. In reality there is no contradiction: In a simple one-component system, to which the validity of the equation d
1623:
1210:
879:
446:
324:
2578:
6609:
1997:
262:
6593:{\displaystyle {\tilde {F}}=\langle H_{0}\rangle _{0}-TS_{0}+\langle \Delta H\rangle _{0}=\langle H\rangle _{0}-TS_{0},}
1877:
10539:
4734:
4492:
3318:
2487:
1394:
1368:
889:
343:
6881:
holds. To see this, consider the difference between the left hand side and the right hand side. We can write this as:
10094:
7351:
295:
9154:{\displaystyle \log \left(P_{r}\right)=\log \left\geq \left\langle -\beta H-\log \left(Z\right)\right\rangle _{r}\,}
3042:
and thus for a system kept at constant temperature and volume and not capable of performing electrical or other non-
8892:
8397:{\displaystyle \left\langle H\right\rangle =\sum _{r}{\tilde {P}}_{r}\left\langle r\left|H\right|r\right\rangle \,}
1751:
1447:
918:
317:
5622:, which is a variational method based on the Bogoliubov inequality. This inequality can be formulated as follows.
3750:
The average internal energy of the system is the expectation value of the energy and can be expressed in terms of
4942:
4801:
1616:
2041:
10529:
10428:
10089:. (Translated from Russian by J. B. Sykes and W. H. Reid) (Third ed.). Boston, MA: Butterworth Heinemann.
2003:
1547:
79:
9856:
6440:
3680:
1442:
3590:
1522:
1295:
272:
10524:
10197:
8716:{\displaystyle \log \left({\tilde {P}}_{r}\right)=-\beta {\tilde {E}}_{r}-\log \left({\tilde {Z}}\right)\,}
7781:
7749:
We can easily generalize this proof to the case of quantum mechanical models. We denote the eigenstates of
6014:
1993:
907:
110:
100:
5732:
where both averages are taken with respect to the canonical distribution defined by the trial
Hamiltonian
5767:
4873:
1868:
115:
105:
10054:
9556:
for the tensors, in which repeated indices in a product are summed. We may integrate the expression for
9199:
7711:
6346:
10305:
10221:
9847:
9443:
6188:
5866:
1399:
1363:
141:
75:
8579:
8133:
7312:
6702:
5470:
for different systems. For example, for a system with a magnetic field or potential, it is true that
2968:
1437:
9931:
4743:
3010:
1192:
940:
386:
199:
189:
9242:
8881:{\displaystyle \left\langle Y\right\rangle _{r}\equiv \left\langle r\left|Y\right|r\right\rangle \,}
2490:
is also valid for a process (without electrical work or composition change) that is not reversible.
10534:
10481:
10464:
10147:
9966:
9559:
2962:
Since the total change in entropy must always be larger or equal to zero, we obtain the inequality
2883:
Since the system is in thermal equilibrium with the heat bath in the initial and the final states,
9413:
9167:
8229:
8200:
8169:
7829:
7752:
7679:
6384:
6214:
6078:
5828:
5735:
5648:
5422:
5048:
10328:
10318:
10244:
5467:
4915:
1852:
1656:
1604:
1432:
1229:
1110:
1055:
1000:
932:
871:
407:
396:
62:
7806:. We denote the diagonal components of the density matrices for the canonical distributions for
5860:
The
Bogoliubov inequality is often applied in the following way. If we write the Hamiltonian as
4462:
834:
787:
702:
655:
567:
520:
10375:
4209:{\displaystyle X=\sum _{r}P_{r}X_{r}={\frac {1}{\beta }}{\frac {\partial \log Z}{\partial x}}.}
1537:
1254:
338:
92:
67:
6971:{\displaystyle \sum _{r}{\tilde {P}}_{r}\log \left({\frac {{\tilde {P}}_{r}}{P_{r}}}\right)\,}
3286:
2532:
2505:
1972:
1949:
1685:
or free enthalpy is most commonly used as a measure of thermodynamic potential (especially in
1457:
738:
603:
10183:
9950:
7202:
where in the last step we have used that both probability distributions are normalized to 1.
4242:
2786:{\displaystyle \Delta S_{\text{bath}}={\frac {Q_{\text{bath}}}{T}}=-{\frac {\Delta U+W}{T}}.}
1705:
1660:
1472:
362:
208:
57:
9969:
for an efficient way to calculate free energy differences and comparison with other methods.
5103:
4288:
2379:
960:
10403:
9471:
8552:
6736:
6675:
6413:
6107:
6051:
5914:
4992:
3632:
3473:
3446:
3259:
1667:
1552:
1477:
1467:
267:
129:
8:
10469:
10448:
9992:
9960:
7940:{\displaystyle P_{r}=\left\langle r\left|{\frac {\exp \left}{Z}}\right|r\right\rangle \,}
5938:
is some exactly solvable
Hamiltonian, then we can apply the above inequality by defining
5019:
1497:
1259:
281:
247:
242:
155:
4265:
1492:
471:
10496:
10408:
10385:
10366:
10346:
10310:
7809:
7659:
5808:
5628:
4222:
3659:
3510:
3509:
A system kept at constant volume, temperature, and particle number is described by the
3498:
3311:
2555:
2483:
1929:
1675:
1586:
1249:
1244:
1197:
813:
766:
681:
634:
546:
499:
429:
413:
300:
252:
237:
227:
36:
30:
8303:{\displaystyle \left\langle {\tilde {H}}\right\rangle =\left\langle H\right\rangle \,}
5416:
These three equations, along with the free energy in terms of the partition function,
10501:
10398:
10371:
10282:
10272:
10265:
10239:
10090:
9927:
9553:
5619:
1698:
1682:
1581:
1542:
1532:
1104:
902:
730:
232:
222:
164:
10393:
10017:
9268:
2873:{\displaystyle \Delta S_{\text{bath}}+\Delta S=-{\frac {\Delta U-T\Delta S+W}{T}}.}
1502:
1487:
1427:
1422:
1239:
1234:
884:
352:
217:
10443:
7345:. If we now substitute in here the expressions for the probability distributions:
2710:
no matter what the system does. Therefore, the entropy change of the heat bath is
10438:
10433:
10361:
10356:
10323:
10250:
10082:
9940:
2284:{\displaystyle \mathrm {d} U=\mathrm {d} (TS)-S\,\mathrm {d} T-p\,\mathrm {d} V,}
1824:
1452:
1300:
954:
595:
418:
179:
146:
10486:
10202:
9956:
9475:
2552:, and the total amount of work that can be extracted, performed by the system,
1644:
1507:
1277:
377:
257:
194:
184:
52:
22:
10518:
10474:
1664:
1576:
894:
463:
424:
136:
10491:
10140:
3433:{\displaystyle dF=-S\,dT-\sum _{i}X_{i}\,dx_{i}+\sum _{j}\mu _{j}\,dN_{j}.}
3046:
work, the total free energy during a spontaneous change can only decrease.
1527:
1512:
1462:
945:
5722:{\displaystyle \left\langle {\tilde {H}}\right\rangle =\langle H\rangle ,}
10300:
10295:
9843:
9193:
are the same then leads to the same conclusion as in the classical case:
2952:{\displaystyle \Delta S_{\text{bath}}+\Delta S=-{\frac {\Delta F+W}{T}}.}
1671:
1482:
290:
10152:
10007:
4939:
is the ground-state degeneracy. The partition function in this limit is
10290:
10078:
5854:
is the free energy of the trial
Hamiltonian. We will prove this below.
5210:
one can find expressions for entropy, pressure and chemical potential:
1694:
1571:
1517:
2366:{\displaystyle \mathrm {d} (U-TS)=-S\,\mathrm {d} T-p\,\mathrm {d} V.}
10351:
10021:
2696:{\displaystyle Q_{\text{bath}}=\Delta U_{\text{bath}}=-(\Delta U+W).}
1859:, in which temperature replaces entropy as the independent variable.
1798:
1686:
169:
10112:"Autoencoders, minimum description length and Helmholtz free energy"
2038:. In case of a reversible change, the work done can be expressed as
10255:
9945:
7524:{\displaystyle {\tilde {P}}_{r}={\frac {\exp \left}{\tilde {Z}}}\,}
4045:, then the generalized force corresponding to an external variable
1285:
1202:
994:
402:
174:
3513:. The probability of finding the system in some energy eigenstate
2493:
10420:
9955:
This page details the
Helmholtz energy from the point of view of
8978:{\displaystyle \log \left\geq \left\langle X\right\rangle _{r}\,}
2526:
2468:{\displaystyle \mathrm {d} F=-S\,\mathrm {d} T-p\,\mathrm {d} V.}
1844:
1725:
391:
9816:
6001:{\displaystyle {\tilde {H}}=H_{0}+\langle \Delta H\rangle _{0}.}
4723:{\displaystyle dU={\frac {1}{\beta }}\,d(\log Z+\beta U)-X\,dx.}
2203:{\displaystyle \mathrm {d} (TS)=T\mathrm {d} S\,+S\mathrm {d} T}
2137:{\displaystyle \mathrm {d} U=T\,\mathrm {d} S-p\,\mathrm {d} V.}
1689:) when it is convenient for applications that occur at constant
10055:"4.3 Entropy, Helmholtz Free Energy and the Partition Function"
3504:
1834:
9989:
Physical memoirs, selected and translated from foreign sources
4647:
This means that the change in the internal energy is given by
3137:
are all fixed. To allow for spontaneous processes at constant
2572:, are well defined quantities. Conservation of energy implies
10087:
Theory of
Elasticity (Course of Theoretical Physics Volume 7)
10013:
9827:
9823:
4795:
This then implies that the entropy of the system is given by
4637:{\displaystyle d(\log Z)=-d(\beta U)+\beta \,dU+\beta X\,dx.}
1806:
9842:
Hinton and Zemel "derive an objective function for training
8895:
for a proof. Taking the logarithm of this inequality gives:
8891:
for the expectation value of the operator Y in the state r.
4108:{\displaystyle X_{r}=-{\frac {\partial E_{r}}{\partial x}}.}
2706:
The heat bath remains in thermal equilibrium at temperature
1847:
of the system (SI: joules per kelvin, CGS: ergs per kelvin).
5551:
5488:
5354:
5291:
5228:
367:
7031:{\displaystyle \log \left(x\right)\geq 1-{\frac {1}{x}}\,}
5092:
3246:{\displaystyle dF=-S\,dT-P\,dV+\sum _{j}\mu _{j}\,dN_{j},}
3158:. The differential of the free energy then generalizes to
1814:
9837:
6175:{\displaystyle \langle X\rangle _{0}=\langle X\rangle .}
9918:"which has exactly the form of Helmholtz free energy".
9542:{\displaystyle \sigma _{ij}=C_{ijkl}\varepsilon _{kl},}
3317:
In case there are other external parameters, the above
1802:
9934:
for thermodynamics history overview and discussion of
4239:. Then changing the system's temperature parameter by
3683:
10077:
9859:
9590:
9562:
9487:
9446:
9416:
9280:
9245:
9202:
9170:
8997:
8904:
8828:
8735:
8621:
8582:
8555:
8416:
8322:
8264:
8232:
8203:
8172:
8136:
7959:
7864:
7832:
7812:
7784:
7755:
7714:
7682:
7662:
7543:
7436:
7354:
7315:
7214:
7050:
6990:
6890:
6748:
6705:
6678:
6667:
6612:
6475:
6443:
6416:
6387:
6349:
6249:
6217:
6191:
6140:
6110:
6081:
6054:
6017:
5947:
5917:
5869:
5831:
5811:
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5738:
5683:
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5479:
5425:
5219:
5146:
5106:
5051:
5022:
4995:
4945:
4918:
4876:
4804:
4746:
4656:
4562:
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4320:
4291:
4268:
4245:
4225:
4127:
4058:
3763:
3662:
3635:
3593:
3527:
3476:
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3289:
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3167:
3013:
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2896:
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2719:
2639:
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2535:
2508:
2420:
2382:
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2219:
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2044:
2006:
1975:
1952:
1932:
1880:
1754:
1113:
1058:
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963:
837:
816:
790:
769:
741:
705:
684:
658:
637:
606:
570:
549:
523:
502:
474:
5825:
is the free energy of the original Hamiltonian, and
3577:{\displaystyle P_{r}={\frac {e^{-\beta E_{r}}}{Z}},}
2619:{\displaystyle \Delta U_{\text{bath}}+\Delta U+W=0.}
1793:
is the Helmholtz free energy (sometimes also called
6657:{\displaystyle F\leq \langle H\rangle _{0}-TS_{0}.}
9907:
9805:
9573:
9541:
9462:
9432:
9399:
9259:
9223:
9185:
9153:
8977:
8880:
8808:
8715:
8604:
8568:
8538:
8396:
8302:
8247:
8218:
8187:
8158:
8119:
7939:
7847:
7818:
7798:
7770:
7735:
7697:
7668:
7645:
7523:
7416:
7337:
7298:
7191:
7030:
6970:
6870:
6727:
6691:
6656:
6592:
6458:
6429:
6402:
6370:
6329:
6232:
6203:
6174:
6123:
6096:
6067:
6036:
6000:
5930:
5900:
5846:
5817:
5794:
5753:
5721:
5666:
5637:
5602:
5455:
5405:
5200:
5138:along with the fundamental thermodynamic relation
5127:
5097:Combining the definition of Helmholtz free energy
5081:
5034:
5008:
4981:
4931:
4904:
4847:
4784:
4722:
4636:
4542:
4478:
4448:
4303:
4277:
4254:
4231:
4219:Suppose that the system has one external variable
4208:
4107:
4030:
3728:
3668:
3648:
3620:
3576:
3489:
3462:
3432:
3302:
3275:
3245:
3031:
2992:
2951:
2872:
2785:
2695:
2618:
2564:
2544:
2517:
2467:
2403:
2365:
2283:
2202:
2136:
2071:
2030:
1984:
1961:
1938:
1916:{\displaystyle \mathrm {d} U=\delta Q\ +\delta W,}
1915:
1778:
1149:
1094:
1039:
984:
846:
822:
799:
775:
750:
714:
690:
667:
643:
618:
579:
555:
532:
508:
483:
10119:Advances in Neural Information Processing Systems
9986:
8726:On the right-hand side we can use the inequality
4543:{\displaystyle U\,d\beta =d(\beta U)-\beta \,dU,}
2147:Applying the product rule for differentiation to
1714:International Union of Pure and Applied Chemistry
10516:
3283:are the numbers of particles of type j and the
3105:is restricted, no process can occur at constant
9234:
7417:{\displaystyle P_{r}={\frac {\exp \left}{Z}}\,}
2494:Minimum free energy and maximum work principles
1837:) of the surroundings, modelled as a heat bath,
9239:In the more general case, the mechanical term
5016:is the ground-state energy. Thus, we see that
4118:The thermal average of this can be written as
3049:This result seems to contradict the equation d
10168:
9817:Application to fundamental equations of state
7309:where the averages are taken with respect to
6307:
6275:
1624:
9470:is the strain tensor. In the case of linear
6626:
6619:
6562:
6555:
6543:
6533:
6505:
6491:
6453:
6444:
6321:
6315:
6302:
6293:
6198:
6192:
6166:
6160:
6148:
6141:
6025:
6018:
5986:
5976:
5713:
5707:
3776:
3770:
3505:Relation to the canonical partition function
1704:The concept of free energy was developed by
10143:and Julio de Paula, Oxford University Press
10109:
9267:must be replaced by the product of volume,
8197:We assume again that the averages of H and
6410:is the free energy of the model defined by
4982:{\displaystyle \Omega _{0}e^{-\beta U_{0}}}
4866:can be determined by considering the limit
4848:{\displaystyle S=k\log Z+{\frac {U}{T}}+c,}
10175:
10161:
2796:The total entropy change is thus given by
2072:{\displaystyle \delta W=-p\,\mathrm {d} V}
1631:
1617:
29:
10182:
9478:, the stress is related to the strain by
9378:
9346:
9321:
9150:
8974:
8877:
8805:
8712:
8535:
8393:
8299:
8116:
7936:
7520:
7413:
7295:
7188:
7027:
6967:
6867:
5201:{\displaystyle dF=-S\,dT-P\,dV+\mu \,dN,}
5188:
5175:
5162:
4772:
4759:
4710:
4676:
4624:
4608:
4530:
4499:
4469:
4436:
4420:
4404:
4368:
3729:{\textstyle Z=\sum _{i}e^{-\beta E_{i}}.}
3413:
3376:
3346:
3226:
3196:
3183:
2453:
2438:
2351:
2336:
2269:
2254:
2185:
2122:
2107:
2060:
2031:{\displaystyle \delta Q=T\,\mathrm {d} S}
2019:
9908:{\displaystyle F=\sum _{i}p_{i}E_{i}-H,}
8612:sum to 1. On the l.h.s. we can replace:
6459:{\displaystyle \langle \Delta H\rangle }
5761:, then the Bogoliubov inequality states
5625:Suppose we replace the real Hamiltonian
5613:
1748:The Helmholtz free energy is defined as
7705:are, by assumption, identical we have:
6048:over the canonical ensemble defined by
5093:Relating free energy to other variables
4870:→ 0. In this limit the entropy becomes
3621:{\displaystyle \beta ={\frac {1}{kT}},}
10517:
10044:" McGraw-Hill: University of Brooklyn.
8819:where we have introduced the notation
1827:of the system (SI: joules, CGS: ergs),
10156:
9838:Application to training auto-encoders
8226:in the canonical ensemble defined by
7799:{\displaystyle \left|r\right\rangle }
6037:{\displaystyle \langle X\rangle _{0}}
1862:
10110:Hinton, G. E.; Zemel, R. S. (1994).
10005:
9164:The fact that the averages of H and
5645:of the model by a trial Hamiltonian
3470:are the external variables, and the
1992:is the work done on the system. The
10146:HyperPhysics Helmholtz Free Energy
5795:{\displaystyle F\leq {\tilde {F}},}
4905:{\displaystyle S=k\log \Omega _{0}}
2079:(ignoring electrical and other non-
13:
10129:
9564:
9380:
9348:
9323:
9282:
9250:
9224:{\displaystyle F\leq {\tilde {F}}}
7736:{\displaystyle F\leq {\tilde {F}}}
6668:Proof of the Bogoliubov inequality
6536:
6447:
6371:{\displaystyle F\leq {\tilde {F}}}
6296:
6131:by a constant, we have in general
5979:
5889:
5568:
5560:
5509:
5501:
5371:
5363:
5312:
5304:
5249:
5241:
4947:
4920:
4893:
4735:fundamental thermodynamic relation
4395:
4381:
4359:
4345:
4194:
4180:
4093:
4078:
4016:
4002:
3945:
3941:
3892:
3888:
3656:is the energy of accessible state
3014:
2981:
2928:
2913:
2897:
2849:
2837:
2822:
2806:
2762:
2720:
2675:
2653:
2598:
2582:
2536:
2509:
2455:
2440:
2422:
2353:
2338:
2305:
2271:
2256:
2232:
2221:
2193:
2178:
2155:
2124:
2109:
2094:
2062:
2021:
1882:
838:
791:
706:
659:
571:
524:
344:Intensive and extensive properties
14:
10551:
10148:Helmholtz and Gibbs Free Energies
9463:{\displaystyle \varepsilon _{ij}}
6240:, as specified above. Therefore,
6204:{\displaystyle \langle X\rangle }
5901:{\displaystyle H=H_{0}+\Delta H,}
1969:is the energy added as heat, and
10211:
9581:to obtain the Helmholtz energy:
8605:{\displaystyle {\tilde {P}}_{r}}
8159:{\displaystyle {\tilde {E}}_{r}}
7338:{\displaystyle {\tilde {P}}_{r}}
7205:We can write the inequality as:
6728:{\displaystyle {\tilde {P}}_{r}}
4733:In the thermodynamic limit, the
2993:{\displaystyle W\leq -\Delta F.}
1600:
1599:
919:Table of thermodynamic equations
10059:theory.physics.manchester.ac.uk
9271:, and an infinitesimal strain:
5542:
5345:
5282:
4862:is some constant. The value of
4785:{\displaystyle dU=T\,dS-X\,dx.}
3032:{\displaystyle \Delta F\leq 0,}
1797:, particularly in the field of
1395:Maxwell's thermodynamic surface
10103:
10071:
10047:
10034:
9999:
9980:
9260:{\displaystyle p\mathrm {d} V}
9215:
9177:
8702:
8671:
8639:
8590:
8499:
8463:
8434:
8354:
8275:
8239:
8210:
8179:
8144:
8109:
8086:
8037:
8020:
7967:
7839:
7762:
7727:
7689:
7586:
7561:
7513:
7485:
7444:
7323:
7237:
7152:
7099:
7068:
6939:
6908:
6831:
6795:
6766:
6713:
6482:
6394:
6362:
6260:
6224:
6104:defined this way differs from
6088:
5954:
5838:
5783:
5745:
5694:
5658:
4701:
4680:
4599:
4590:
4578:
4566:
4521:
4512:
4336:
4324:
2687:
2672:
2324:
2309:
2245:
2236:
2168:
2159:
1716:(IUPAC) recommends the symbol
1129:
1117:
1074:
1062:
1019:
1007:
979:
967:
1:
9973:
9574:{\displaystyle \mathrm {d} F}
4262:and the external variable by
1833:is the absolute temperature (
1779:{\displaystyle F\equiv U-TS,}
1743:
1732:is also used in reference to
1296:Mechanical equivalent of heat
10198:Principle of maximum entropy
9433:{\displaystyle \sigma _{ij}}
9235:Generalized Helmholtz energy
9186:{\displaystyle {\tilde {H}}}
8248:{\displaystyle {\tilde {H}}}
8219:{\displaystyle {\tilde {H}}}
8188:{\displaystyle {\tilde {H}}}
7848:{\displaystyle {\tilde {H}}}
7771:{\displaystyle {\tilde {H}}}
7698:{\displaystyle {\tilde {H}}}
6403:{\displaystyle {\tilde {F}}}
6233:{\displaystyle {\tilde {H}}}
6097:{\displaystyle {\tilde {H}}}
5847:{\displaystyle {\tilde {F}}}
5754:{\displaystyle {\tilde {H}}}
5667:{\displaystyle {\tilde {H}}}
5456:{\displaystyle F=-kT\log Z,}
5082:{\displaystyle F=-kT\log Z.}
2411:enables to rewrite this as
1994:second law of thermodynamics
1871:in a closed system provides
1851:The Helmholtz energy is the
908:Onsager reciprocal relations
7:
9921:
4932:{\displaystyle \Omega _{0}}
1869:first law of thermodynamics
1400:Entropy as energy dispersal
1211:"Perpetual motion" machines
1150:{\displaystyle G(T,p)=H-TS}
1095:{\displaystyle A(T,V)=U-TS}
1040:{\displaystyle H(S,p)=U+pV}
10:
10556:
10222:Statistical thermodynamics
10006:Gold, Victor, ed. (2019).
9987:von Helmholtz, H. (1882).
9848:minimum description length
9440:is the stress tensor, and
6211:is still the average over
4479:{\displaystyle U\,d\beta }
4041:If the system is in state
3154:of particles of each type
3113:, since there is a unique
847:{\displaystyle \partial T}
800:{\displaystyle \partial V}
715:{\displaystyle \partial p}
668:{\displaystyle \partial V}
580:{\displaystyle \partial T}
533:{\displaystyle \partial S}
10540:Thermodynamic free energy
10457:
10419:
10384:
10339:
10281:
10220:
10209:
10190:
10040:Levine, Ira. N. (1978). "
9932:thermodynamic free energy
8988:This allows us to write:
4285:will lead to a change in
3077:constant seems to imply d
1659:that measures the useful
1321:An Inquiry Concerning the
10482:Condensed matter physics
10465:Statistical field theory
9967:Bennett acceptance ratio
8549:still holds as both the
6340:and thus the inequality
5468:Legendre transformations
3303:{\displaystyle \mu _{j}}
2545:{\displaystyle \Delta S}
2518:{\displaystyle \Delta U}
1985:{\displaystyle \delta W}
1962:{\displaystyle \delta Q}
1946:is the internal energy,
1334:Heterogeneous Substances
751:{\displaystyle \alpha =}
619:{\displaystyle \beta =-}
10340:Mathematical approaches
10329:Lennard-Jones potential
10245:thermodynamic potential
9552:where we are now using
8166:are the eigenvalues of
6381:holds. The free energy
4255:{\displaystyle d\beta }
3321:further generalizes to
1855:of the internal energy
1853:Legendre transformation
1657:thermodynamic potential
16:Thermodynamic potential
10376:conformal field theory
9909:
9807:
9575:
9543:
9464:
9434:
9401:
9261:
9225:
9187:
9155:
8979:
8882:
8810:
8717:
8606:
8570:
8540:
8398:
8304:
8249:
8220:
8189:
8160:
8121:
7941:
7849:
7820:
7800:
7772:
7737:
7699:
7670:
7656:Since the averages of
7647:
7525:
7418:
7339:
7300:
7193:
7032:
6972:
6872:
6729:
6693:
6658:
6594:
6460:
6431:
6404:
6372:
6331:
6234:
6205:
6176:
6125:
6098:
6069:
6038:
6002:
5932:
5902:
5848:
5819:
5796:
5755:
5723:
5668:
5639:
5604:
5457:
5407:
5202:
5129:
5128:{\displaystyle F=U-TS}
5083:
5036:
5010:
4983:
4933:
4906:
4849:
4786:
4724:
4638:
4544:
4480:
4450:
4305:
4304:{\displaystyle \log Z}
4279:
4256:
4233:
4210:
4109:
4032:
3730:
3670:
3650:
3622:
3578:
3491:
3464:
3434:
3310:are the corresponding
3304:
3277:
3247:
3033:
2994:
2953:
2874:
2787:
2697:
2620:
2566:
2546:
2519:
2469:
2405:
2404:{\displaystyle F=U-TS}
2367:
2285:
2204:
2138:
2073:
2032:
1986:
1963:
1940:
1917:
1780:
1151:
1096:
1041:
986:
985:{\displaystyle U(S,V)}
848:
824:
801:
777:
752:
716:
692:
669:
645:
620:
581:
557:
534:
510:
485:
464:Specific heat capacity
68:Quantum thermodynamics
10530:Hermann von Helmholtz
10291:Ferromagnetism models
10184:Statistical mechanics
9951:Statistical mechanics
9910:
9808:
9576:
9544:
9465:
9435:
9402:
9262:
9226:
9188:
9156:
8980:
8883:
8811:
8718:
8607:
8571:
8569:{\displaystyle P_{r}}
8541:
8399:
8305:
8250:
8221:
8190:
8161:
8122:
7942:
7850:
7821:
7801:
7773:
7738:
7700:
7671:
7648:
7526:
7419:
7340:
7301:
7194:
7033:
6973:
6873:
6735:, respectively. From
6730:
6694:
6692:{\displaystyle P_{r}}
6659:
6595:
6461:
6432:
6430:{\displaystyle H_{0}}
6405:
6373:
6332:
6235:
6206:
6177:
6126:
6124:{\displaystyle H_{0}}
6099:
6070:
6068:{\displaystyle H_{0}}
6044:to be the average of
6039:
6011:Here we have defined
6003:
5933:
5931:{\displaystyle H_{0}}
5903:
5849:
5820:
5797:
5756:
5724:
5669:
5640:
5614:Bogoliubov inequality
5605:
5458:
5408:
5203:
5130:
5084:
5037:
5011:
5009:{\displaystyle U_{0}}
4984:
4934:
4907:
4850:
4787:
4725:
4639:
4545:
4481:
4451:
4306:
4280:
4257:
4234:
4211:
4110:
4033:
3731:
3671:
3651:
3649:{\displaystyle E_{r}}
3623:
3579:
3517:, for any microstate
3492:
3490:{\displaystyle X_{i}}
3465:
3463:{\displaystyle x_{i}}
3435:
3305:
3278:
3276:{\displaystyle N_{j}}
3248:
3125:) relation, and thus
3034:
2995:
2954:
2875:
2788:
2698:
2621:
2567:
2547:
2520:
2470:
2406:
2368:
2286:
2205:
2139:
2074:
2033:
1987:
1964:
1941:
1918:
1781:
1706:Hermann von Helmholtz
1649:Helmholtz free energy
1332:On the Equilibrium of
1152:
1097:
1050:Helmholtz free energy
1042:
987:
849:
825:
802:
778:
753:
717:
693:
670:
646:
621:
582:
558:
535:
511:
486:
9993:Taylor & Francis
9857:
9588:
9560:
9485:
9474:materials that obey
9444:
9414:
9278:
9243:
9200:
9168:
8995:
8902:
8826:
8733:
8619:
8580:
8553:
8414:
8320:
8262:
8230:
8201:
8170:
8134:
7957:
7862:
7830:
7810:
7782:
7753:
7712:
7680:
7660:
7541:
7434:
7352:
7313:
7212:
7048:
6988:
6888:
6746:
6703:
6676:
6610:
6473:
6441:
6414:
6385:
6347:
6247:
6215:
6189:
6138:
6108:
6079:
6052:
6015:
5945:
5915:
5867:
5829:
5809:
5768:
5736:
5681:
5649:
5629:
5477:
5423:
5217:
5144:
5104:
5049:
5020:
4993:
4943:
4916:
4874:
4802:
4744:
4654:
4560:
4493:
4463:
4318:
4289:
4266:
4243:
4223:
4125:
4056:
3761:
3681:
3660:
3633:
3591:
3525:
3474:
3447:
3328:
3287:
3260:
3165:
3011:
2969:
2894:
2803:
2717:
2637:
2579:
2556:
2533:
2506:
2418:
2380:
2301:
2217:
2151:
2090:
2042:
2004:
1973:
1950:
1930:
1878:
1752:
1701:of pure substances.
1668:thermodynamic system
1345:Motive Power of Fire
1111:
1056:
1001:
961:
913:Bridgman's equations
890:Fundamental relation
835:
814:
788:
767:
739:
703:
682:
656:
635:
604:
568:
547:
521:
500:
472:
10525:Physical quantities
10470:elementary particle
10235:partition functions
9961:statistical physics
5035:{\displaystyle c=0}
3312:chemical potentials
2482:is a thermodynamic
1693:. For example, in
1323:Source ... Friction
1255:Loschmidt's paradox
447:Material properties
325:Conjugate variables
10497:information theory
10404:correlation length
10399:Critical exponents
10386:Critical phenomena
10367:stochastic process
10347:Boltzmann equation
10240:equations of state
10139:, 7th edition, by
10137:Physical Chemistry
10042:Physical Chemistry
9905:
9875:
9826:, as given by the
9803:
9801:
9775:
9684:
9571:
9539:
9460:
9430:
9397:
9367:
9307:
9257:
9221:
9183:
9151:
8975:
8878:
8806:
8713:
8602:
8566:
8536:
8491:
8426:
8394:
8346:
8300:
8245:
8216:
8185:
8156:
8117:
7937:
7855:in this basis as:
7845:
7816:
7796:
7768:
7733:
7695:
7666:
7643:
7521:
7414:
7335:
7296:
7189:
7139:
7060:
7028:
6968:
6900:
6868:
6823:
6758:
6725:
6689:
6654:
6590:
6466:. This means that
6456:
6427:
6400:
6368:
6327:
6230:
6201:
6172:
6121:
6094:
6065:
6034:
5998:
5928:
5898:
5844:
5815:
5792:
5751:
5719:
5664:
5635:
5600:
5453:
5403:
5198:
5125:
5079:
5032:
5006:
4979:
4929:
4902:
4845:
4782:
4720:
4634:
4540:
4476:
4446:
4301:
4278:{\displaystyle dx}
4275:
4252:
4229:
4206:
4143:
4105:
4028:
3963:
3879:
3824:
3791:
3744:partition function
3726:
3699:
3666:
3646:
3618:
3574:
3511:canonical ensemble
3499:generalized forces
3497:the corresponding
3487:
3460:
3430:
3402:
3365:
3300:
3273:
3243:
3215:
3029:
2990:
2949:
2870:
2783:
2693:
2616:
2562:
2542:
2515:
2465:
2401:
2376:The definition of
2363:
2281:
2200:
2134:
2069:
2028:
1998:reversible process
1982:
1959:
1936:
1913:
1863:Formal development
1776:
1738:Helmholtz function
1699:equations of state
1663:obtainable from a
1587:Order and disorder
1343:Reflections on the
1250:Heat death paradox
1147:
1092:
1037:
982:
844:
820:
797:
773:
748:
712:
688:
665:
641:
616:
577:
553:
530:
506:
484:{\displaystyle c=}
481:
454:Property databases
430:Reduced properties
414:Chemical potential
378:Functions of state
301:Thermal efficiency
37:Carnot heat engine
10512:
10511:
10502:Boltzmann machine
10372:mean-field theory
10273:Maxwell relations
9928:Gibbs free energy
9866:
9766:
9723:
9675:
9613:
9554:Einstein notation
9358:
9295:
9218:
9180:
8705:
8674:
8642:
8593:
8502:
8482:
8466:
8437:
8417:
8357:
8337:
8278:
8242:
8213:
8182:
8147:
8114:
8112:
8089:
8042:
8040:
8023:
7970:
7922:
7842:
7819:{\displaystyle H}
7765:
7730:
7692:
7669:{\displaystyle H}
7589:
7564:
7518:
7516:
7488:
7447:
7411:
7326:
7240:
7155:
7130:
7121:
7102:
7071:
7051:
7041:it follows that:
7025:
6961:
6942:
6911:
6891:
6834:
6814:
6798:
6769:
6749:
6737:Gibbs' inequality
6716:
6485:
6397:
6365:
6263:
6227:
6091:
5957:
5841:
5818:{\displaystyle F}
5786:
5748:
5697:
5661:
5638:{\displaystyle H}
5620:mean-field theory
5575:
5516:
5378:
5319:
5256:
4834:
4674:
4402:
4366:
4232:{\displaystyle x}
4201:
4175:
4134:
4100:
4023:
3991:
3954:
3952:
3928:
3899:
3870:
3865:
3815:
3782:
3690:
3669:{\displaystyle r}
3613:
3569:
3393:
3356:
3206:
2944:
2907:
2865:
2816:
2778:
2751:
2745:
2730:
2663:
2647:
2592:
2565:{\displaystyle W}
2484:function of state
1939:{\displaystyle U}
1900:
1683:Gibbs free energy
1681:In contrast, the
1641:
1640:
1582:Self-organization
1407:
1406:
1105:Gibbs free energy
903:Maxwell relations
861:
860:
857:
856:
823:{\displaystyle V}
776:{\displaystyle 1}
731:Thermal expansion
725:
724:
691:{\displaystyle V}
644:{\displaystyle 1}
590:
589:
556:{\displaystyle N}
509:{\displaystyle T}
437:
436:
353:Process functions
339:Property diagrams
318:System properties
308:
307:
273:Endoreversibility
165:Equation of state
10547:
10394:Phase transition
10215:
10214:
10177:
10170:
10163:
10154:
10153:
10123:
10122:
10116:
10107:
10101:
10100:
10075:
10069:
10068:
10066:
10065:
10051:
10045:
10038:
10032:
10031:
10029:
10028:
10022:10.1351/goldbook
10003:
9997:
9996:
9984:
9914:
9912:
9911:
9906:
9895:
9894:
9885:
9884:
9874:
9812:
9810:
9809:
9804:
9802:
9795:
9794:
9785:
9784:
9774:
9753:
9752:
9740:
9739:
9724:
9716:
9708:
9704:
9703:
9694:
9693:
9683:
9662:
9661:
9649:
9648:
9636:
9635:
9614:
9606:
9580:
9578:
9577:
9572:
9567:
9548:
9546:
9545:
9540:
9535:
9534:
9522:
9521:
9500:
9499:
9469:
9467:
9466:
9461:
9459:
9458:
9439:
9437:
9436:
9431:
9429:
9428:
9406:
9404:
9403:
9398:
9393:
9392:
9383:
9377:
9376:
9366:
9351:
9339:
9338:
9326:
9320:
9319:
9306:
9285:
9266:
9264:
9263:
9258:
9253:
9230:
9228:
9227:
9222:
9220:
9219:
9211:
9192:
9190:
9189:
9184:
9182:
9181:
9173:
9160:
9158:
9157:
9152:
9149:
9148:
9143:
9139:
9138:
9100:
9096:
9095:
9090:
9086:
9085:
9081:
9080:
9021:
9017:
9016:
8984:
8982:
8981:
8976:
8973:
8972:
8967:
8952:
8948:
8947:
8942:
8938:
8937:
8887:
8885:
8884:
8879:
8876:
8872:
8868:
8846:
8845:
8840:
8815:
8813:
8812:
8807:
8804:
8800:
8799:
8794:
8769:
8768:
8763:
8759:
8758:
8722:
8720:
8719:
8714:
8711:
8707:
8706:
8698:
8682:
8681:
8676:
8675:
8667:
8654:
8650:
8649:
8644:
8643:
8635:
8611:
8609:
8608:
8603:
8601:
8600:
8595:
8594:
8586:
8575:
8573:
8572:
8567:
8565:
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8542:
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8534:
8530:
8529:
8510:
8509:
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8490:
8478:
8474:
8473:
8468:
8467:
8459:
8445:
8444:
8439:
8438:
8430:
8425:
8403:
8401:
8400:
8395:
8392:
8388:
8384:
8365:
8364:
8359:
8358:
8350:
8345:
8333:
8309:
8307:
8306:
8301:
8298:
8284:
8280:
8279:
8271:
8254:
8252:
8251:
8246:
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8243:
8235:
8225:
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8222:
8217:
8215:
8214:
8206:
8194:
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8191:
8186:
8184:
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8098:
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8033:
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7995:
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7935:
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7917:
7913:
7891:
7874:
7873:
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7844:
7843:
7835:
7825:
7823:
7822:
7817:
7805:
7803:
7802:
7797:
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7769:
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7766:
7758:
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7723:
7704:
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7652:
7650:
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7642:
7638:
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7582:
7566:
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7557:
7530:
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7517:
7509:
7507:
7506:
7502:
7501:
7490:
7489:
7481:
7460:
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7440:
7423:
7421:
7420:
7415:
7412:
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7364:
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6757:
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6311:
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6289:
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6156:
6155:
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6035:
6033:
6032:
6007:
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5904:
5899:
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5779:
5760:
5758:
5757:
5752:
5750:
5749:
5741:
5728:
5726:
5725:
5720:
5703:
5699:
5698:
5690:
5673:
5671:
5670:
5665:
5663:
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5642:
5641:
5636:
5609:
5607:
5606:
5601:
5596:
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5526:
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5507:
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5387:
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5369:
5361:
5341:
5340:
5329:
5325:
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5310:
5302:
5278:
5277:
5266:
5262:
5261:
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5255:
5247:
5239:
5207:
5205:
5204:
5199:
5134:
5132:
5131:
5126:
5088:
5086:
5085:
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5041:
5039:
5038:
5033:
5015:
5013:
5012:
5007:
5005:
5004:
4988:
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4985:
4980:
4978:
4977:
4976:
4975:
4955:
4954:
4938:
4936:
4935:
4930:
4928:
4927:
4911:
4909:
4908:
4903:
4901:
4900:
4854:
4852:
4851:
4846:
4835:
4827:
4791:
4789:
4788:
4783:
4729:
4727:
4726:
4721:
4675:
4667:
4643:
4641:
4640:
4635:
4549:
4547:
4546:
4541:
4485:
4483:
4482:
4477:
4455:
4453:
4452:
4447:
4403:
4401:
4393:
4379:
4367:
4365:
4357:
4343:
4310:
4308:
4307:
4302:
4284:
4282:
4281:
4276:
4261:
4259:
4258:
4253:
4238:
4236:
4235:
4230:
4215:
4213:
4212:
4207:
4202:
4200:
4192:
4178:
4176:
4168:
4163:
4162:
4153:
4152:
4142:
4114:
4112:
4111:
4106:
4101:
4099:
4091:
4090:
4089:
4076:
4068:
4067:
4037:
4035:
4034:
4029:
4024:
4022:
4014:
4000:
3992:
3987:
3986:
3985:
3984:
3983:
3962:
3953:
3951:
3940:
3934:
3929:
3924:
3923:
3922:
3921:
3920:
3900:
3898:
3887:
3881:
3878:
3866:
3861:
3860:
3859:
3850:
3849:
3848:
3847:
3826:
3823:
3811:
3810:
3801:
3800:
3790:
3735:
3733:
3732:
3727:
3722:
3721:
3720:
3719:
3698:
3675:
3673:
3672:
3667:
3655:
3653:
3652:
3647:
3645:
3644:
3627:
3625:
3624:
3619:
3614:
3612:
3601:
3583:
3581:
3580:
3575:
3570:
3565:
3564:
3563:
3562:
3542:
3537:
3536:
3496:
3494:
3493:
3488:
3486:
3485:
3469:
3467:
3466:
3461:
3459:
3458:
3439:
3437:
3436:
3431:
3426:
3425:
3412:
3411:
3401:
3389:
3388:
3375:
3374:
3364:
3309:
3307:
3306:
3301:
3299:
3298:
3282:
3280:
3279:
3274:
3272:
3271:
3252:
3250:
3249:
3244:
3239:
3238:
3225:
3224:
3214:
3038:
3036:
3035:
3030:
2999:
2997:
2996:
2991:
2958:
2956:
2955:
2950:
2945:
2940:
2926:
2909:
2908:
2905:
2879:
2877:
2876:
2871:
2866:
2861:
2835:
2818:
2817:
2814:
2792:
2790:
2789:
2784:
2779:
2774:
2760:
2752:
2747:
2746:
2743:
2737:
2732:
2731:
2728:
2702:
2700:
2699:
2694:
2665:
2664:
2661:
2649:
2648:
2645:
2625:
2623:
2622:
2617:
2594:
2593:
2590:
2571:
2569:
2568:
2563:
2551:
2549:
2548:
2543:
2524:
2522:
2521:
2516:
2474:
2472:
2471:
2466:
2458:
2443:
2425:
2410:
2408:
2407:
2402:
2372:
2370:
2369:
2364:
2356:
2341:
2308:
2290:
2288:
2287:
2282:
2274:
2259:
2235:
2224:
2209:
2207:
2206:
2201:
2196:
2181:
2158:
2143:
2141:
2140:
2135:
2127:
2112:
2097:
2078:
2076:
2075:
2070:
2065:
2037:
2035:
2034:
2029:
2024:
1991:
1989:
1988:
1983:
1968:
1966:
1965:
1960:
1945:
1943:
1942:
1937:
1922:
1920:
1919:
1914:
1898:
1885:
1785:
1783:
1782:
1777:
1722:Helmholtz energy
1653:Helmholtz energy
1633:
1626:
1619:
1603:
1602:
1310:Key publications
1291:
1290:("living force")
1240:Brownian ratchet
1235:Entropy and life
1230:Entropy and time
1181:
1180:
1156:
1154:
1153:
1148:
1101:
1099:
1098:
1093:
1046:
1044:
1043:
1038:
991:
989:
988:
983:
885:Clausius theorem
880:Carnot's theorem
853:
851:
850:
845:
829:
827:
826:
821:
806:
804:
803:
798:
782:
780:
779:
774:
761:
760:
757:
755:
754:
749:
721:
719:
718:
713:
697:
695:
694:
689:
674:
672:
671:
666:
650:
648:
647:
642:
629:
628:
625:
623:
622:
617:
586:
584:
583:
578:
562:
560:
559:
554:
539:
537:
536:
531:
515:
513:
512:
507:
494:
493:
490:
488:
487:
482:
460:
459:
333:
332:
152:
151:
33:
19:
18:
10555:
10554:
10550:
10549:
10548:
10546:
10545:
10544:
10535:State functions
10515:
10514:
10513:
10508:
10453:
10415:
10380:
10362:BBGKY hierarchy
10357:Vlasov equation
10335:
10324:depletion force
10317:Particles with
10277:
10216:
10212:
10207:
10186:
10181:
10132:
10130:Further reading
10127:
10126:
10114:
10108:
10104:
10097:
10083:Lifshitz, E. M.
10076:
10072:
10063:
10061:
10053:
10052:
10048:
10039:
10035:
10026:
10024:
10004:
10000:
9985:
9981:
9976:
9941:Grand potential
9924:
9890:
9886:
9880:
9876:
9870:
9858:
9855:
9854:
9840:
9819:
9800:
9799:
9790:
9786:
9780:
9776:
9770:
9745:
9741:
9732:
9728:
9715:
9706:
9705:
9699:
9695:
9689:
9685:
9679:
9654:
9650:
9641:
9637:
9622:
9618:
9605:
9598:
9591:
9589:
9586:
9585:
9563:
9561:
9558:
9557:
9527:
9523:
9508:
9504:
9492:
9488:
9486:
9483:
9482:
9451:
9447:
9445:
9442:
9441:
9421:
9417:
9415:
9412:
9411:
9388:
9384:
9379:
9372:
9368:
9362:
9347:
9331:
9327:
9322:
9312:
9308:
9299:
9281:
9279:
9276:
9275:
9249:
9244:
9241:
9240:
9237:
9210:
9209:
9201:
9198:
9197:
9172:
9171:
9169:
9166:
9165:
9144:
9128:
9109:
9105:
9104:
9091:
9070:
9051:
9047:
9040:
9036:
9035:
9031:
9012:
9008:
9004:
8996:
8993:
8992:
8968:
8957:
8956:
8943:
8927:
8920:
8916:
8915:
8911:
8903:
8900:
8899:
8858:
8854:
8850:
8841:
8830:
8829:
8827:
8824:
8823:
8795:
8784:
8783:
8779:
8764:
8748:
8741:
8737:
8736:
8734:
8731:
8730:
8697:
8696:
8692:
8677:
8666:
8665:
8664:
8645:
8634:
8633:
8632:
8628:
8620:
8617:
8616:
8596:
8585:
8584:
8583:
8581:
8578:
8577:
8560:
8556:
8554:
8551:
8550:
8525:
8521:
8517:
8505:
8494:
8493:
8492:
8486:
8469:
8458:
8457:
8456:
8452:
8440:
8429:
8428:
8427:
8421:
8415:
8412:
8411:
8407:The inequality
8374:
8370:
8366:
8360:
8349:
8348:
8347:
8341:
8323:
8321:
8318:
8317:
8288:
8270:
8269:
8265:
8263:
8260:
8259:
8234:
8233:
8231:
8228:
8227:
8205:
8204:
8202:
8199:
8198:
8174:
8173:
8171:
8168:
8167:
8150:
8139:
8138:
8137:
8135:
8132:
8131:
8104:
8092:
8081:
8080:
8079:
8072:
8068:
8061:
8059:
8032:
8015:
8014:
8007:
8003:
7996:
7994:
7990:
7986:
7982:
7973:
7962:
7961:
7960:
7958:
7955:
7954:
7903:
7899:
7892:
7890:
7886:
7882:
7878:
7869:
7865:
7863:
7860:
7859:
7834:
7833:
7831:
7828:
7827:
7811:
7808:
7807:
7785:
7783:
7780:
7779:
7757:
7756:
7754:
7751:
7750:
7722:
7721:
7713:
7710:
7709:
7684:
7683:
7681:
7678:
7677:
7661:
7658:
7657:
7627:
7608:
7604:
7581:
7580:
7576:
7556:
7555:
7548:
7544:
7542:
7539:
7538:
7508:
7491:
7480:
7479:
7472:
7468:
7461:
7459:
7450:
7439:
7438:
7437:
7435:
7432:
7431:
7391:
7381:
7377:
7370:
7368:
7359:
7355:
7353:
7350:
7349:
7329:
7318:
7317:
7316:
7314:
7311:
7310:
7280:
7276:
7272:
7265:
7261:
7243:
7232:
7231:
7230:
7226:
7219:
7215:
7213:
7210:
7209:
7171:
7167:
7158:
7147:
7146:
7145:
7144:
7140:
7134:
7115:
7111:
7105:
7094:
7093:
7092:
7090:
7086:
7074:
7063:
7062:
7061:
7055:
7049:
7046:
7045:
7017:
6997:
6989:
6986:
6985:
6955:
6951:
6945:
6934:
6933:
6932:
6930:
6926:
6914:
6903:
6902:
6901:
6895:
6889:
6886:
6885:
6857:
6853:
6849:
6837:
6826:
6825:
6824:
6818:
6801:
6790:
6789:
6788:
6784:
6772:
6761:
6760:
6759:
6753:
6747:
6744:
6743:
6719:
6708:
6707:
6706:
6704:
6701:
6700:
6683:
6679:
6677:
6674:
6673:
6670:
6645:
6641:
6629:
6625:
6611:
6608:
6607:
6581:
6577:
6565:
6561:
6546:
6542:
6524:
6520:
6508:
6504:
6498:
6494:
6477:
6476:
6474:
6471:
6470:
6442:
6439:
6438:
6421:
6417:
6415:
6412:
6411:
6389:
6388:
6386:
6383:
6382:
6357:
6356:
6348:
6345:
6344:
6306:
6305:
6284:
6280:
6274:
6273:
6255:
6254:
6250:
6248:
6245:
6244:
6219:
6218:
6216:
6213:
6212:
6190:
6187:
6186:
6151:
6147:
6139:
6136:
6135:
6115:
6111:
6109:
6106:
6105:
6083:
6082:
6080:
6077:
6076:
6059:
6055:
6053:
6050:
6049:
6028:
6024:
6016:
6013:
6012:
5989:
5985:
5967:
5963:
5949:
5948:
5946:
5943:
5942:
5922:
5918:
5916:
5913:
5912:
5880:
5876:
5868:
5865:
5864:
5833:
5832:
5830:
5827:
5826:
5810:
5807:
5806:
5778:
5777:
5769:
5766:
5765:
5740:
5739:
5737:
5734:
5733:
5689:
5688:
5684:
5682:
5679:
5678:
5653:
5652:
5650:
5647:
5646:
5630:
5627:
5626:
5616:
5585:
5567:
5559:
5557:
5553:
5550:
5549:
5527:
5508:
5500:
5498:
5494:
5490:
5487:
5486:
5478:
5475:
5474:
5424:
5421:
5420:
5388:
5370:
5362:
5360:
5356:
5353:
5352:
5330:
5311:
5303:
5301:
5297:
5293:
5290:
5289:
5267:
5248:
5240:
5238:
5234:
5230:
5227:
5226:
5218:
5215:
5214:
5145:
5142:
5141:
5105:
5102:
5101:
5095:
5050:
5047:
5046:
5021:
5018:
5017:
5000:
4996:
4994:
4991:
4990:
4971:
4967:
4960:
4956:
4950:
4946:
4944:
4941:
4940:
4923:
4919:
4917:
4914:
4913:
4896:
4892:
4875:
4872:
4871:
4826:
4803:
4800:
4799:
4745:
4742:
4741:
4666:
4655:
4652:
4651:
4561:
4558:
4557:
4494:
4491:
4490:
4464:
4461:
4460:
4394:
4380:
4378:
4358:
4344:
4342:
4319:
4316:
4315:
4290:
4287:
4286:
4267:
4264:
4263:
4244:
4241:
4240:
4224:
4221:
4220:
4193:
4179:
4177:
4167:
4158:
4154:
4148:
4144:
4138:
4126:
4123:
4122:
4092:
4085:
4081:
4077:
4075:
4063:
4059:
4057:
4054:
4053:
4015:
4001:
3999:
3979:
3975:
3968:
3964:
3958:
3944:
3939:
3935:
3933:
3916:
3912:
3905:
3901:
3891:
3886:
3882:
3880:
3874:
3855:
3851:
3843:
3839:
3832:
3828:
3827:
3825:
3819:
3806:
3802:
3796:
3792:
3786:
3762:
3759:
3758:
3715:
3711:
3704:
3700:
3694:
3682:
3679:
3678:
3661:
3658:
3657:
3640:
3636:
3634:
3631:
3630:
3605:
3600:
3592:
3589:
3588:
3558:
3554:
3547:
3543:
3541:
3532:
3528:
3526:
3523:
3522:
3507:
3481:
3477:
3475:
3472:
3471:
3454:
3450:
3448:
3445:
3444:
3421:
3417:
3407:
3403:
3397:
3384:
3380:
3370:
3366:
3360:
3329:
3326:
3325:
3294:
3290:
3288:
3285:
3284:
3267:
3263:
3261:
3258:
3257:
3234:
3230:
3220:
3216:
3210:
3166:
3163:
3162:
3153:
3081:= 0, and hence
3012:
3009:
3008:
2970:
2967:
2966:
2927:
2925:
2904:
2900:
2895:
2892:
2891:
2836:
2834:
2813:
2809:
2804:
2801:
2800:
2761:
2759:
2742:
2738:
2736:
2727:
2723:
2718:
2715:
2714:
2660:
2656:
2644:
2640:
2638:
2635:
2634:
2589:
2585:
2580:
2577:
2576:
2557:
2554:
2553:
2534:
2531:
2530:
2507:
2504:
2503:
2496:
2454:
2439:
2421:
2419:
2416:
2415:
2381:
2378:
2377:
2352:
2337:
2304:
2302:
2299:
2298:
2270:
2255:
2231:
2220:
2218:
2215:
2214:
2192:
2177:
2154:
2152:
2149:
2148:
2123:
2108:
2093:
2091:
2088:
2087:
2061:
2043:
2040:
2039:
2020:
2005:
2002:
2001:
1974:
1971:
1970:
1951:
1948:
1947:
1931:
1928:
1927:
1881:
1879:
1876:
1875:
1865:
1825:internal energy
1753:
1750:
1749:
1746:
1637:
1592:
1591:
1567:
1559:
1558:
1557:
1417:
1409:
1408:
1387:
1373:
1348:
1344:
1337:
1333:
1326:
1322:
1289:
1282:
1264:
1245:Maxwell's demon
1207:
1178:
1177:
1161:
1160:
1159:
1112:
1109:
1108:
1107:
1057:
1054:
1053:
1052:
1002:
999:
998:
997:
962:
959:
958:
957:
955:Internal energy
950:
935:
925:
924:
899:
874:
864:
863:
862:
836:
833:
832:
815:
812:
811:
789:
786:
785:
768:
765:
764:
740:
737:
736:
704:
701:
700:
683:
680:
679:
657:
654:
653:
636:
633:
632:
605:
602:
601:
596:Compressibility
569:
566:
565:
548:
545:
544:
522:
519:
518:
501:
498:
497:
473:
470:
469:
449:
439:
438:
419:Particle number
372:
331:
320:
310:
309:
268:Irreversibility
180:State of matter
147:Isolated system
132:
122:
121:
120:
95:
85:
84:
80:Non-equilibrium
72:
47:
39:
17:
12:
11:
5:
10553:
10543:
10542:
10537:
10532:
10527:
10510:
10509:
10507:
10506:
10505:
10504:
10499:
10494:
10487:Complex system
10484:
10479:
10478:
10477:
10472:
10461:
10459:
10455:
10454:
10452:
10451:
10446:
10441:
10436:
10431:
10425:
10423:
10417:
10416:
10414:
10413:
10412:
10411:
10406:
10396:
10390:
10388:
10382:
10381:
10379:
10378:
10369:
10364:
10359:
10354:
10349:
10343:
10341:
10337:
10336:
10334:
10333:
10332:
10331:
10326:
10315:
10314:
10313:
10308:
10303:
10298:
10287:
10285:
10279:
10278:
10276:
10275:
10270:
10269:
10268:
10263:
10258:
10253:
10242:
10237:
10232:
10226:
10224:
10218:
10217:
10210:
10208:
10206:
10205:
10203:ergodic theory
10200:
10194:
10192:
10188:
10187:
10180:
10179:
10172:
10165:
10157:
10151:
10150:
10144:
10131:
10128:
10125:
10124:
10102:
10095:
10070:
10046:
10033:
9998:
9978:
9977:
9975:
9972:
9971:
9970:
9964:
9953:
9948:
9943:
9938:
9923:
9920:
9916:
9915:
9904:
9901:
9898:
9893:
9889:
9883:
9879:
9873:
9869:
9865:
9862:
9839:
9836:
9818:
9815:
9814:
9813:
9798:
9793:
9789:
9783:
9779:
9773:
9769:
9765:
9762:
9759:
9756:
9751:
9748:
9744:
9738:
9735:
9731:
9727:
9722:
9719:
9714:
9711:
9709:
9707:
9702:
9698:
9692:
9688:
9682:
9678:
9674:
9671:
9668:
9665:
9660:
9657:
9653:
9647:
9644:
9640:
9634:
9631:
9628:
9625:
9621:
9617:
9612:
9609:
9604:
9601:
9599:
9597:
9594:
9593:
9570:
9566:
9550:
9549:
9538:
9533:
9530:
9526:
9520:
9517:
9514:
9511:
9507:
9503:
9498:
9495:
9491:
9457:
9454:
9450:
9427:
9424:
9420:
9408:
9407:
9396:
9391:
9387:
9382:
9375:
9371:
9365:
9361:
9357:
9354:
9350:
9345:
9342:
9337:
9334:
9330:
9325:
9318:
9315:
9311:
9305:
9302:
9298:
9294:
9291:
9288:
9284:
9256:
9252:
9248:
9236:
9233:
9232:
9231:
9217:
9214:
9208:
9205:
9179:
9176:
9162:
9161:
9147:
9142:
9137:
9134:
9131:
9127:
9124:
9121:
9118:
9115:
9112:
9108:
9103:
9099:
9094:
9089:
9084:
9079:
9076:
9073:
9069:
9066:
9063:
9060:
9057:
9054:
9050:
9046:
9043:
9039:
9034:
9030:
9027:
9024:
9020:
9015:
9011:
9007:
9003:
9000:
8986:
8985:
8971:
8966:
8963:
8960:
8955:
8951:
8946:
8941:
8936:
8933:
8930:
8926:
8923:
8919:
8914:
8910:
8907:
8889:
8888:
8875:
8871:
8867:
8864:
8861:
8857:
8853:
8849:
8844:
8839:
8836:
8833:
8817:
8816:
8803:
8798:
8793:
8790:
8787:
8782:
8778:
8775:
8772:
8767:
8762:
8757:
8754:
8751:
8747:
8744:
8740:
8724:
8723:
8710:
8704:
8701:
8695:
8691:
8688:
8685:
8680:
8673:
8670:
8663:
8660:
8657:
8653:
8648:
8641:
8638:
8631:
8627:
8624:
8599:
8592:
8589:
8563:
8559:
8547:
8546:
8533:
8528:
8524:
8520:
8516:
8513:
8508:
8501:
8498:
8489:
8485:
8481:
8477:
8472:
8465:
8462:
8455:
8451:
8448:
8443:
8436:
8433:
8424:
8420:
8405:
8404:
8391:
8387:
8383:
8380:
8377:
8373:
8369:
8363:
8356:
8353:
8344:
8340:
8336:
8332:
8329:
8326:
8311:
8310:
8297:
8294:
8291:
8287:
8283:
8277:
8274:
8268:
8255:are the same:
8241:
8238:
8212:
8209:
8181:
8178:
8153:
8146:
8143:
8128:
8127:
8111:
8108:
8101:
8095:
8088:
8085:
8078:
8075:
8071:
8067:
8064:
8058:
8054:
8050:
8046:
8039:
8036:
8029:
8022:
8019:
8013:
8010:
8006:
8002:
7999:
7993:
7989:
7985:
7981:
7976:
7969:
7966:
7948:
7947:
7934:
7930:
7926:
7921:
7916:
7912:
7909:
7906:
7902:
7898:
7895:
7889:
7885:
7881:
7877:
7872:
7868:
7841:
7838:
7815:
7794:
7791:
7788:
7764:
7761:
7744:
7743:
7729:
7726:
7720:
7717:
7691:
7688:
7665:
7654:
7653:
7641:
7636:
7633:
7630:
7626:
7623:
7620:
7617:
7614:
7611:
7607:
7603:
7599:
7594:
7588:
7585:
7579:
7575:
7572:
7569:
7563:
7560:
7554:
7551:
7547:
7532:
7531:
7515:
7512:
7505:
7500:
7497:
7494:
7487:
7484:
7478:
7475:
7471:
7467:
7464:
7458:
7453:
7446:
7443:
7425:
7424:
7410:
7405:
7400:
7397:
7394:
7390:
7387:
7384:
7380:
7376:
7373:
7367:
7362:
7358:
7332:
7325:
7322:
7307:
7306:
7293:
7288:
7283:
7279:
7275:
7271:
7268:
7264:
7260:
7256:
7251:
7246:
7239:
7236:
7229:
7225:
7222:
7218:
7200:
7199:
7187:
7184:
7180:
7174:
7170:
7166:
7161:
7154:
7151:
7143:
7137:
7133:
7129:
7125:
7118:
7114:
7108:
7101:
7098:
7089:
7085:
7082:
7077:
7070:
7067:
7058:
7054:
7039:
7038:
7024:
7021:
7016:
7013:
7010:
7006:
7003:
7000:
6996:
6993:
6979:
6978:
6965:
6958:
6954:
6948:
6941:
6938:
6929:
6925:
6922:
6917:
6910:
6907:
6898:
6894:
6879:
6878:
6865:
6860:
6856:
6852:
6848:
6845:
6840:
6833:
6830:
6821:
6817:
6813:
6809:
6804:
6797:
6794:
6787:
6783:
6780:
6775:
6768:
6765:
6756:
6752:
6739:we know that:
6722:
6715:
6712:
6686:
6682:
6669:
6666:
6665:
6664:
6653:
6648:
6644:
6640:
6637:
6632:
6628:
6624:
6621:
6618:
6615:
6601:
6600:
6589:
6584:
6580:
6576:
6573:
6568:
6564:
6560:
6557:
6554:
6549:
6545:
6541:
6538:
6535:
6532:
6527:
6523:
6519:
6516:
6511:
6507:
6501:
6497:
6493:
6490:
6484:
6481:
6455:
6452:
6449:
6446:
6424:
6420:
6396:
6393:
6379:
6378:
6364:
6361:
6355:
6352:
6338:
6337:
6326:
6323:
6320:
6317:
6314:
6309:
6304:
6301:
6298:
6295:
6292:
6287:
6283:
6277:
6272:
6268:
6262:
6259:
6253:
6226:
6223:
6200:
6197:
6194:
6183:
6182:
6171:
6168:
6165:
6162:
6159:
6154:
6150:
6146:
6143:
6118:
6114:
6090:
6087:
6062:
6058:
6031:
6027:
6023:
6020:
6009:
6008:
5997:
5992:
5988:
5984:
5981:
5978:
5975:
5970:
5966:
5962:
5956:
5953:
5925:
5921:
5909:
5908:
5897:
5894:
5891:
5888:
5883:
5879:
5875:
5872:
5840:
5837:
5814:
5803:
5802:
5791:
5785:
5782:
5776:
5773:
5747:
5744:
5730:
5729:
5718:
5715:
5712:
5709:
5706:
5702:
5696:
5693:
5687:
5660:
5657:
5634:
5615:
5612:
5611:
5610:
5599:
5594:
5591:
5588:
5583:
5579:
5573:
5570:
5565:
5562:
5556:
5552:
5548:
5545:
5541:
5536:
5533:
5530:
5525:
5520:
5514:
5511:
5506:
5503:
5497:
5493:
5489:
5485:
5482:
5464:
5463:
5452:
5449:
5446:
5443:
5440:
5437:
5434:
5431:
5428:
5414:
5413:
5402:
5397:
5394:
5391:
5386:
5382:
5376:
5373:
5368:
5365:
5359:
5355:
5351:
5348:
5344:
5339:
5336:
5333:
5328:
5323:
5317:
5314:
5309:
5306:
5300:
5296:
5292:
5288:
5285:
5281:
5276:
5273:
5270:
5265:
5260:
5254:
5251:
5246:
5243:
5237:
5233:
5229:
5225:
5222:
5197:
5194:
5191:
5187:
5184:
5181:
5178:
5174:
5171:
5168:
5165:
5161:
5158:
5155:
5152:
5149:
5136:
5135:
5124:
5121:
5118:
5115:
5112:
5109:
5094:
5091:
5090:
5089:
5078:
5075:
5072:
5069:
5066:
5063:
5060:
5057:
5054:
5031:
5028:
5025:
5003:
4999:
4974:
4970:
4966:
4963:
4959:
4953:
4949:
4926:
4922:
4899:
4895:
4891:
4888:
4885:
4882:
4879:
4856:
4855:
4844:
4841:
4838:
4833:
4830:
4825:
4822:
4819:
4816:
4813:
4810:
4807:
4793:
4792:
4781:
4778:
4775:
4771:
4768:
4765:
4762:
4758:
4755:
4752:
4749:
4731:
4730:
4719:
4716:
4713:
4709:
4706:
4703:
4700:
4697:
4694:
4691:
4688:
4685:
4682:
4679:
4673:
4670:
4665:
4662:
4659:
4645:
4644:
4633:
4630:
4627:
4623:
4620:
4617:
4614:
4611:
4607:
4604:
4601:
4598:
4595:
4592:
4589:
4586:
4583:
4580:
4577:
4574:
4571:
4568:
4565:
4551:
4550:
4539:
4536:
4533:
4529:
4526:
4523:
4520:
4517:
4514:
4511:
4508:
4505:
4502:
4498:
4475:
4472:
4468:
4457:
4456:
4445:
4442:
4439:
4435:
4432:
4429:
4426:
4423:
4419:
4416:
4413:
4410:
4407:
4400:
4397:
4392:
4389:
4386:
4383:
4377:
4374:
4371:
4364:
4361:
4356:
4353:
4350:
4347:
4341:
4338:
4335:
4332:
4329:
4326:
4323:
4300:
4297:
4294:
4274:
4271:
4251:
4248:
4228:
4217:
4216:
4205:
4199:
4196:
4191:
4188:
4185:
4182:
4174:
4171:
4166:
4161:
4157:
4151:
4147:
4141:
4137:
4133:
4130:
4116:
4115:
4104:
4098:
4095:
4088:
4084:
4080:
4074:
4071:
4066:
4062:
4039:
4038:
4027:
4021:
4018:
4013:
4010:
4007:
4004:
3998:
3995:
3990:
3982:
3978:
3974:
3971:
3967:
3961:
3957:
3950:
3947:
3943:
3938:
3932:
3927:
3919:
3915:
3911:
3908:
3904:
3897:
3894:
3890:
3885:
3877:
3873:
3869:
3864:
3858:
3854:
3846:
3842:
3838:
3835:
3831:
3822:
3818:
3814:
3809:
3805:
3799:
3795:
3789:
3785:
3781:
3778:
3775:
3772:
3769:
3766:
3742:is called the
3737:
3736:
3725:
3718:
3714:
3710:
3707:
3703:
3697:
3693:
3689:
3686:
3676:
3665:
3643:
3639:
3628:
3617:
3611:
3608:
3604:
3599:
3596:
3573:
3568:
3561:
3557:
3553:
3550:
3546:
3540:
3535:
3531:
3521:, is given by
3506:
3503:
3484:
3480:
3457:
3453:
3441:
3440:
3429:
3424:
3420:
3416:
3410:
3406:
3400:
3396:
3392:
3387:
3383:
3379:
3373:
3369:
3363:
3359:
3355:
3352:
3349:
3345:
3342:
3339:
3336:
3333:
3297:
3293:
3270:
3266:
3254:
3253:
3242:
3237:
3233:
3229:
3223:
3219:
3213:
3209:
3205:
3202:
3199:
3195:
3192:
3189:
3186:
3182:
3179:
3176:
3173:
3170:
3149:
3040:
3039:
3028:
3025:
3022:
3019:
3016:
3001:
3000:
2989:
2986:
2983:
2980:
2977:
2974:
2960:
2959:
2948:
2943:
2939:
2936:
2933:
2930:
2924:
2921:
2918:
2915:
2912:
2903:
2899:
2881:
2880:
2869:
2864:
2860:
2857:
2854:
2851:
2848:
2845:
2842:
2839:
2833:
2830:
2827:
2824:
2821:
2812:
2808:
2794:
2793:
2782:
2777:
2773:
2770:
2767:
2764:
2758:
2755:
2750:
2741:
2735:
2726:
2722:
2704:
2703:
2692:
2689:
2686:
2683:
2680:
2677:
2674:
2671:
2668:
2659:
2655:
2652:
2643:
2627:
2626:
2615:
2612:
2609:
2606:
2603:
2600:
2597:
2588:
2584:
2561:
2541:
2538:
2514:
2511:
2495:
2492:
2476:
2475:
2464:
2461:
2457:
2452:
2449:
2446:
2442:
2437:
2434:
2431:
2428:
2424:
2400:
2397:
2394:
2391:
2388:
2385:
2374:
2373:
2362:
2359:
2355:
2350:
2347:
2344:
2340:
2335:
2332:
2329:
2326:
2323:
2320:
2317:
2314:
2311:
2307:
2292:
2291:
2280:
2277:
2273:
2268:
2265:
2262:
2258:
2253:
2250:
2247:
2244:
2241:
2238:
2234:
2230:
2227:
2223:
2199:
2195:
2191:
2188:
2184:
2180:
2176:
2173:
2170:
2167:
2164:
2161:
2157:
2145:
2144:
2133:
2130:
2126:
2121:
2118:
2115:
2111:
2106:
2103:
2100:
2096:
2083:work) and so:
2068:
2064:
2059:
2056:
2053:
2050:
2047:
2027:
2023:
2018:
2015:
2012:
2009:
1981:
1978:
1958:
1955:
1935:
1924:
1923:
1912:
1909:
1906:
1903:
1897:
1894:
1891:
1888:
1884:
1864:
1861:
1849:
1848:
1838:
1828:
1818:
1775:
1772:
1769:
1766:
1763:
1760:
1757:
1745:
1742:
1670:at a constant
1645:thermodynamics
1639:
1638:
1636:
1635:
1628:
1621:
1613:
1610:
1609:
1608:
1607:
1594:
1593:
1590:
1589:
1584:
1579:
1574:
1568:
1565:
1564:
1561:
1560:
1556:
1555:
1550:
1545:
1540:
1535:
1530:
1525:
1520:
1515:
1510:
1505:
1500:
1495:
1490:
1485:
1480:
1475:
1470:
1465:
1460:
1455:
1450:
1445:
1440:
1435:
1430:
1425:
1419:
1418:
1415:
1414:
1411:
1410:
1405:
1404:
1403:
1402:
1397:
1389:
1388:
1386:
1385:
1382:
1378:
1375:
1374:
1372:
1371:
1366:
1364:Thermodynamics
1360:
1357:
1356:
1352:
1351:
1350:
1349:
1340:
1338:
1329:
1327:
1318:
1313:
1312:
1306:
1305:
1304:
1303:
1298:
1293:
1281:
1280:
1278:Caloric theory
1274:
1271:
1270:
1266:
1265:
1263:
1262:
1257:
1252:
1247:
1242:
1237:
1232:
1226:
1223:
1222:
1216:
1215:
1214:
1213:
1206:
1205:
1200:
1195:
1189:
1186:
1185:
1179:
1176:
1175:
1172:
1168:
1167:
1166:
1163:
1162:
1158:
1157:
1146:
1143:
1140:
1137:
1134:
1131:
1128:
1125:
1122:
1119:
1116:
1102:
1091:
1088:
1085:
1082:
1079:
1076:
1073:
1070:
1067:
1064:
1061:
1047:
1036:
1033:
1030:
1027:
1024:
1021:
1018:
1015:
1012:
1009:
1006:
992:
981:
978:
975:
972:
969:
966:
951:
949:
948:
943:
937:
936:
931:
930:
927:
926:
923:
922:
915:
910:
905:
898:
897:
892:
887:
882:
876:
875:
870:
869:
866:
865:
859:
858:
855:
854:
843:
840:
830:
819:
808:
807:
796:
793:
783:
772:
758:
747:
744:
734:
727:
726:
723:
722:
711:
708:
698:
687:
676:
675:
664:
661:
651:
640:
626:
615:
612:
609:
599:
592:
591:
588:
587:
576:
573:
563:
552:
541:
540:
529:
526:
516:
505:
491:
480:
477:
467:
458:
457:
456:
450:
445:
444:
441:
440:
435:
434:
433:
432:
427:
422:
411:
400:
381:
380:
374:
373:
371:
370:
365:
359:
356:
355:
349:
348:
347:
346:
341:
322:
321:
316:
315:
312:
311:
306:
305:
304:
303:
298:
293:
285:
284:
278:
277:
276:
275:
270:
265:
260:
258:Free expansion
255:
250:
245:
240:
235:
230:
225:
220:
212:
211:
205:
204:
203:
202:
197:
195:Control volume
192:
187:
185:Phase (matter)
182:
177:
172:
167:
159:
158:
150:
149:
144:
139:
133:
128:
127:
124:
123:
119:
118:
113:
108:
103:
97:
96:
91:
90:
87:
86:
83:
82:
71:
70:
65:
60:
55:
49:
48:
45:
44:
41:
40:
35:The classical
34:
26:
25:
23:Thermodynamics
15:
9:
6:
4:
3:
2:
10552:
10541:
10538:
10536:
10533:
10531:
10528:
10526:
10523:
10522:
10520:
10503:
10500:
10498:
10495:
10493:
10490:
10489:
10488:
10485:
10483:
10480:
10476:
10475:superfluidity
10473:
10471:
10468:
10467:
10466:
10463:
10462:
10460:
10456:
10450:
10447:
10445:
10442:
10440:
10437:
10435:
10432:
10430:
10427:
10426:
10424:
10422:
10418:
10410:
10407:
10405:
10402:
10401:
10400:
10397:
10395:
10392:
10391:
10389:
10387:
10383:
10377:
10373:
10370:
10368:
10365:
10363:
10360:
10358:
10355:
10353:
10350:
10348:
10345:
10344:
10342:
10338:
10330:
10327:
10325:
10322:
10321:
10320:
10316:
10312:
10309:
10307:
10304:
10302:
10299:
10297:
10294:
10293:
10292:
10289:
10288:
10286:
10284:
10280:
10274:
10271:
10267:
10264:
10262:
10259:
10257:
10254:
10252:
10249:
10248:
10246:
10243:
10241:
10238:
10236:
10233:
10231:
10228:
10227:
10225:
10223:
10219:
10204:
10201:
10199:
10196:
10195:
10193:
10189:
10185:
10178:
10173:
10171:
10166:
10164:
10159:
10158:
10155:
10149:
10145:
10142:
10138:
10134:
10133:
10120:
10113:
10106:
10098:
10096:0-7506-2633-X
10092:
10088:
10084:
10080:
10079:Landau, L. D.
10074:
10060:
10056:
10050:
10043:
10037:
10023:
10019:
10015:
10011:
10010:
10002:
9994:
9990:
9983:
9979:
9968:
9965:
9962:
9958:
9954:
9952:
9949:
9947:
9944:
9942:
9939:
9937:
9933:
9929:
9926:
9925:
9919:
9902:
9899:
9896:
9891:
9887:
9881:
9877:
9871:
9867:
9863:
9860:
9853:
9852:
9851:
9849:
9846:based on the
9845:
9835:
9833:
9829:
9825:
9796:
9791:
9787:
9781:
9777:
9771:
9767:
9763:
9760:
9757:
9754:
9749:
9746:
9742:
9736:
9733:
9729:
9725:
9720:
9717:
9712:
9710:
9700:
9696:
9690:
9686:
9680:
9676:
9672:
9669:
9666:
9663:
9658:
9655:
9651:
9645:
9642:
9638:
9632:
9629:
9626:
9623:
9619:
9615:
9610:
9607:
9602:
9600:
9595:
9584:
9583:
9582:
9568:
9555:
9536:
9531:
9528:
9524:
9518:
9515:
9512:
9509:
9505:
9501:
9496:
9493:
9489:
9481:
9480:
9479:
9477:
9473:
9455:
9452:
9448:
9425:
9422:
9418:
9394:
9389:
9385:
9373:
9369:
9363:
9359:
9355:
9352:
9343:
9340:
9335:
9332:
9328:
9316:
9313:
9309:
9303:
9300:
9296:
9292:
9289:
9286:
9274:
9273:
9272:
9270:
9254:
9246:
9212:
9206:
9203:
9196:
9195:
9194:
9174:
9145:
9140:
9135:
9132:
9129:
9125:
9122:
9119:
9116:
9113:
9110:
9106:
9101:
9097:
9092:
9087:
9082:
9077:
9074:
9071:
9067:
9064:
9061:
9058:
9055:
9052:
9048:
9044:
9041:
9037:
9032:
9028:
9025:
9022:
9018:
9013:
9009:
9005:
9001:
8998:
8991:
8990:
8989:
8969:
8964:
8961:
8958:
8953:
8949:
8944:
8939:
8934:
8931:
8928:
8924:
8921:
8917:
8912:
8908:
8905:
8898:
8897:
8896:
8894:
8873:
8869:
8865:
8862:
8859:
8855:
8851:
8847:
8842:
8837:
8834:
8831:
8822:
8821:
8820:
8801:
8796:
8791:
8788:
8785:
8780:
8776:
8773:
8770:
8765:
8760:
8755:
8752:
8749:
8745:
8742:
8738:
8729:
8728:
8727:
8708:
8699:
8693:
8689:
8686:
8683:
8678:
8668:
8661:
8658:
8655:
8651:
8646:
8636:
8629:
8625:
8622:
8615:
8614:
8613:
8597:
8587:
8561:
8557:
8531:
8526:
8522:
8518:
8514:
8511:
8506:
8496:
8487:
8483:
8479:
8475:
8470:
8460:
8453:
8449:
8446:
8441:
8431:
8422:
8418:
8410:
8409:
8408:
8389:
8385:
8381:
8378:
8375:
8371:
8367:
8361:
8351:
8342:
8338:
8334:
8330:
8327:
8324:
8316:
8315:
8314:
8295:
8292:
8289:
8285:
8281:
8272:
8266:
8258:
8257:
8256:
8236:
8207:
8195:
8176:
8151:
8141:
8106:
8099:
8093:
8083:
8076:
8073:
8069:
8065:
8062:
8056:
8052:
8048:
8044:
8034:
8027:
8017:
8011:
8008:
8004:
8000:
7997:
7991:
7987:
7983:
7979:
7974:
7964:
7953:
7952:
7951:
7932:
7928:
7924:
7919:
7914:
7910:
7907:
7904:
7900:
7896:
7893:
7887:
7883:
7879:
7875:
7870:
7866:
7858:
7857:
7856:
7836:
7813:
7792:
7789:
7786:
7759:
7747:
7724:
7718:
7715:
7708:
7707:
7706:
7686:
7663:
7639:
7634:
7631:
7628:
7624:
7621:
7618:
7615:
7612:
7609:
7605:
7601:
7597:
7592:
7583:
7577:
7573:
7570:
7567:
7558:
7552:
7549:
7545:
7537:
7536:
7535:
7510:
7503:
7498:
7495:
7492:
7482:
7476:
7473:
7469:
7465:
7462:
7456:
7451:
7441:
7430:
7429:
7428:
7408:
7403:
7398:
7395:
7392:
7388:
7385:
7382:
7378:
7374:
7371:
7365:
7360:
7356:
7348:
7347:
7346:
7330:
7320:
7291:
7286:
7281:
7277:
7273:
7269:
7266:
7262:
7258:
7254:
7249:
7244:
7234:
7227:
7223:
7220:
7216:
7208:
7207:
7206:
7203:
7185:
7182:
7178:
7172:
7168:
7164:
7159:
7149:
7141:
7135:
7131:
7127:
7123:
7116:
7112:
7106:
7096:
7087:
7083:
7080:
7075:
7065:
7056:
7052:
7044:
7043:
7042:
7022:
7019:
7014:
7011:
7008:
7004:
7001:
6998:
6994:
6991:
6984:
6983:
6982:
6963:
6956:
6952:
6946:
6936:
6927:
6923:
6920:
6915:
6905:
6896:
6892:
6884:
6883:
6882:
6863:
6858:
6854:
6850:
6846:
6843:
6838:
6828:
6819:
6815:
6811:
6807:
6802:
6792:
6785:
6781:
6778:
6773:
6763:
6754:
6750:
6742:
6741:
6740:
6738:
6720:
6710:
6684:
6680:
6651:
6646:
6642:
6638:
6635:
6630:
6622:
6616:
6613:
6606:
6605:
6604:
6587:
6582:
6578:
6574:
6571:
6566:
6558:
6552:
6547:
6539:
6530:
6525:
6521:
6517:
6514:
6509:
6499:
6495:
6488:
6479:
6469:
6468:
6467:
6450:
6422:
6418:
6391:
6359:
6353:
6350:
6343:
6342:
6341:
6324:
6318:
6312:
6299:
6290:
6285:
6281:
6270:
6266:
6257:
6251:
6243:
6242:
6241:
6221:
6195:
6169:
6163:
6157:
6152:
6144:
6134:
6133:
6132:
6116:
6112:
6085:
6060:
6056:
6047:
6029:
6021:
5995:
5990:
5982:
5973:
5968:
5964:
5960:
5951:
5941:
5940:
5939:
5923:
5919:
5895:
5892:
5886:
5881:
5877:
5873:
5870:
5863:
5862:
5861:
5858:
5855:
5835:
5812:
5789:
5780:
5774:
5771:
5764:
5763:
5762:
5742:
5716:
5710:
5704:
5700:
5691:
5685:
5677:
5676:
5675:
5655:
5632:
5623:
5621:
5597:
5592:
5589:
5586:
5581:
5577:
5571:
5563:
5554:
5546:
5543:
5539:
5534:
5531:
5528:
5523:
5518:
5512:
5504:
5495:
5491:
5483:
5480:
5473:
5472:
5471:
5469:
5450:
5447:
5444:
5441:
5438:
5435:
5432:
5429:
5426:
5419:
5418:
5417:
5400:
5395:
5392:
5389:
5384:
5380:
5374:
5366:
5357:
5349:
5346:
5342:
5337:
5334:
5331:
5326:
5321:
5315:
5307:
5298:
5294:
5286:
5283:
5279:
5274:
5271:
5268:
5263:
5258:
5252:
5244:
5235:
5231:
5223:
5220:
5213:
5212:
5211:
5208:
5195:
5192:
5189:
5185:
5182:
5179:
5176:
5172:
5169:
5166:
5163:
5159:
5156:
5153:
5150:
5147:
5139:
5122:
5119:
5116:
5113:
5110:
5107:
5100:
5099:
5098:
5076:
5073:
5070:
5067:
5064:
5061:
5058:
5055:
5052:
5045:
5044:
5043:
5029:
5026:
5023:
5001:
4997:
4972:
4968:
4964:
4961:
4957:
4951:
4924:
4897:
4889:
4886:
4883:
4880:
4877:
4869:
4865:
4861:
4842:
4839:
4836:
4831:
4828:
4823:
4820:
4817:
4814:
4811:
4808:
4805:
4798:
4797:
4796:
4779:
4776:
4773:
4769:
4766:
4763:
4760:
4756:
4753:
4750:
4747:
4740:
4739:
4738:
4737:should hold:
4736:
4717:
4714:
4711:
4707:
4704:
4698:
4695:
4692:
4689:
4686:
4683:
4677:
4671:
4668:
4663:
4660:
4657:
4650:
4649:
4648:
4631:
4628:
4625:
4621:
4618:
4615:
4612:
4609:
4605:
4602:
4596:
4593:
4587:
4584:
4581:
4575:
4572:
4569:
4563:
4556:
4555:
4554:
4537:
4534:
4531:
4527:
4524:
4518:
4515:
4509:
4506:
4503:
4500:
4496:
4489:
4488:
4487:
4473:
4470:
4466:
4443:
4440:
4437:
4433:
4430:
4427:
4424:
4421:
4417:
4414:
4411:
4408:
4405:
4398:
4390:
4387:
4384:
4375:
4372:
4369:
4362:
4354:
4351:
4348:
4339:
4333:
4330:
4327:
4321:
4314:
4313:
4312:
4298:
4295:
4292:
4272:
4269:
4249:
4246:
4226:
4203:
4197:
4189:
4186:
4183:
4172:
4169:
4164:
4159:
4155:
4149:
4145:
4139:
4135:
4131:
4128:
4121:
4120:
4119:
4102:
4096:
4086:
4082:
4072:
4069:
4064:
4060:
4052:
4051:
4050:
4048:
4044:
4025:
4019:
4011:
4008:
4005:
3996:
3993:
3988:
3980:
3976:
3972:
3969:
3965:
3959:
3955:
3948:
3936:
3930:
3925:
3917:
3913:
3909:
3906:
3902:
3895:
3883:
3875:
3871:
3867:
3862:
3856:
3852:
3844:
3840:
3836:
3833:
3829:
3820:
3816:
3812:
3807:
3803:
3797:
3793:
3787:
3783:
3779:
3773:
3767:
3764:
3757:
3756:
3755:
3753:
3748:
3745:
3741:
3723:
3716:
3712:
3708:
3705:
3701:
3695:
3691:
3687:
3684:
3677:
3663:
3641:
3637:
3629:
3615:
3609:
3606:
3602:
3597:
3594:
3587:
3586:
3585:
3571:
3566:
3559:
3555:
3551:
3548:
3544:
3538:
3533:
3529:
3520:
3516:
3512:
3502:
3500:
3482:
3478:
3455:
3451:
3427:
3422:
3418:
3414:
3408:
3404:
3398:
3394:
3390:
3385:
3381:
3377:
3371:
3367:
3361:
3357:
3353:
3350:
3347:
3343:
3340:
3337:
3334:
3331:
3324:
3323:
3322:
3320:
3315:
3313:
3295:
3291:
3268:
3264:
3240:
3235:
3231:
3227:
3221:
3217:
3211:
3207:
3203:
3200:
3197:
3193:
3190:
3187:
3184:
3180:
3177:
3174:
3171:
3168:
3161:
3160:
3159:
3157:
3152:
3148:
3144:
3140:
3136:
3132:
3128:
3124:
3120:
3116:
3112:
3108:
3104:
3100:
3096:
3092:
3088:
3084:
3080:
3076:
3072:
3069:, as keeping
3068:
3064:
3060:
3056:
3052:
3047:
3045:
3026:
3023:
3020:
3017:
3007:
3006:
3005:
2987:
2984:
2978:
2975:
2972:
2965:
2964:
2963:
2946:
2941:
2937:
2934:
2931:
2922:
2919:
2916:
2910:
2901:
2890:
2889:
2888:
2886:
2867:
2862:
2858:
2855:
2852:
2846:
2843:
2840:
2831:
2828:
2825:
2819:
2810:
2799:
2798:
2797:
2780:
2775:
2771:
2768:
2765:
2756:
2753:
2748:
2739:
2733:
2724:
2713:
2712:
2711:
2709:
2690:
2684:
2681:
2678:
2669:
2666:
2657:
2650:
2641:
2633:
2632:
2631:
2613:
2610:
2607:
2604:
2601:
2595:
2586:
2575:
2574:
2573:
2559:
2539:
2528:
2512:
2500:
2491:
2489:
2485:
2481:
2462:
2459:
2450:
2447:
2444:
2435:
2432:
2429:
2426:
2414:
2413:
2412:
2398:
2395:
2392:
2389:
2386:
2383:
2360:
2357:
2348:
2345:
2342:
2333:
2330:
2327:
2321:
2318:
2315:
2312:
2297:
2296:
2295:
2278:
2275:
2266:
2263:
2260:
2251:
2248:
2242:
2239:
2228:
2225:
2213:
2212:
2211:
2210:, it follows
2197:
2189:
2186:
2182:
2174:
2171:
2165:
2162:
2131:
2128:
2119:
2116:
2113:
2104:
2101:
2098:
2086:
2085:
2084:
2082:
2066:
2057:
2054:
2051:
2048:
2045:
2025:
2016:
2013:
2010:
2007:
1999:
1995:
1979:
1976:
1956:
1953:
1933:
1910:
1907:
1904:
1901:
1895:
1892:
1889:
1886:
1874:
1873:
1872:
1870:
1860:
1858:
1854:
1846:
1842:
1839:
1836:
1832:
1829:
1826:
1822:
1819:
1816:
1812:
1808:
1804:
1800:
1796:
1792:
1789:
1788:
1787:
1773:
1770:
1767:
1764:
1761:
1758:
1755:
1741:
1739:
1735:
1731:
1728:, the symbol
1727:
1723:
1720:and the name
1719:
1715:
1711:
1707:
1702:
1700:
1696:
1692:
1688:
1684:
1679:
1677:
1673:
1669:
1666:
1662:
1658:
1654:
1650:
1646:
1634:
1629:
1627:
1622:
1620:
1615:
1614:
1612:
1611:
1606:
1598:
1597:
1596:
1595:
1588:
1585:
1583:
1580:
1578:
1577:Self-assembly
1575:
1573:
1570:
1569:
1563:
1562:
1554:
1551:
1549:
1548:van der Waals
1546:
1544:
1541:
1539:
1536:
1534:
1531:
1529:
1526:
1524:
1521:
1519:
1516:
1514:
1511:
1509:
1506:
1504:
1501:
1499:
1496:
1494:
1491:
1489:
1486:
1484:
1481:
1479:
1476:
1474:
1473:von Helmholtz
1471:
1469:
1466:
1464:
1461:
1459:
1456:
1454:
1451:
1449:
1446:
1444:
1441:
1439:
1436:
1434:
1431:
1429:
1426:
1424:
1421:
1420:
1413:
1412:
1401:
1398:
1396:
1393:
1392:
1391:
1390:
1383:
1380:
1379:
1377:
1376:
1370:
1367:
1365:
1362:
1361:
1359:
1358:
1354:
1353:
1347:
1346:
1339:
1336:
1335:
1328:
1325:
1324:
1317:
1316:
1315:
1314:
1311:
1308:
1307:
1302:
1299:
1297:
1294:
1292:
1288:
1284:
1283:
1279:
1276:
1275:
1273:
1272:
1268:
1267:
1261:
1258:
1256:
1253:
1251:
1248:
1246:
1243:
1241:
1238:
1236:
1233:
1231:
1228:
1227:
1225:
1224:
1221:
1218:
1217:
1212:
1209:
1208:
1204:
1201:
1199:
1196:
1194:
1191:
1190:
1188:
1187:
1183:
1182:
1173:
1170:
1169:
1165:
1164:
1144:
1141:
1138:
1135:
1132:
1126:
1123:
1120:
1114:
1106:
1103:
1089:
1086:
1083:
1080:
1077:
1071:
1068:
1065:
1059:
1051:
1048:
1034:
1031:
1028:
1025:
1022:
1016:
1013:
1010:
1004:
996:
993:
976:
973:
970:
964:
956:
953:
952:
947:
944:
942:
939:
938:
934:
929:
928:
921:
920:
916:
914:
911:
909:
906:
904:
901:
900:
896:
895:Ideal gas law
893:
891:
888:
886:
883:
881:
878:
877:
873:
868:
867:
841:
831:
817:
810:
809:
794:
784:
770:
763:
762:
759:
745:
742:
735:
732:
729:
728:
709:
699:
685:
678:
677:
662:
652:
638:
631:
630:
627:
613:
610:
607:
600:
597:
594:
593:
574:
564:
550:
543:
542:
527:
517:
503:
496:
495:
492:
478:
475:
468:
465:
462:
461:
455:
452:
451:
448:
443:
442:
431:
428:
426:
425:Vapor quality
423:
421:
420:
415:
412:
410:
409:
404:
401:
398:
394:
393:
388:
385:
384:
383:
382:
379:
376:
375:
369:
366:
364:
361:
360:
358:
357:
354:
351:
350:
345:
342:
340:
337:
336:
335:
334:
330:
326:
319:
314:
313:
302:
299:
297:
294:
292:
289:
288:
287:
286:
283:
280:
279:
274:
271:
269:
266:
264:
263:Reversibility
261:
259:
256:
254:
251:
249:
246:
244:
241:
239:
236:
234:
231:
229:
226:
224:
221:
219:
216:
215:
214:
213:
210:
207:
206:
201:
198:
196:
193:
191:
188:
186:
183:
181:
178:
176:
173:
171:
168:
166:
163:
162:
161:
160:
157:
154:
153:
148:
145:
143:
140:
138:
137:Closed system
135:
134:
131:
126:
125:
117:
114:
112:
109:
107:
104:
102:
99:
98:
94:
89:
88:
81:
77:
74:
73:
69:
66:
64:
61:
59:
56:
54:
51:
50:
43:
42:
38:
32:
28:
27:
24:
21:
20:
10458:Applications
10409:size scaling
10260:
10141:Peter Atkins
10136:
10118:
10105:
10086:
10073:
10062:. Retrieved
10058:
10049:
10041:
10036:
10025:. Retrieved
10008:
10001:
9988:
9982:
9935:
9917:
9844:auto-encoder
9841:
9820:
9551:
9409:
9238:
9163:
8987:
8890:
8818:
8725:
8548:
8406:
8312:
8196:
8129:
7949:
7748:
7745:
7655:
7533:
7426:
7308:
7204:
7201:
7040:
6980:
6880:
6671:
6602:
6380:
6339:
6184:
6045:
6010:
5910:
5859:
5856:
5804:
5731:
5624:
5617:
5465:
5415:
5209:
5140:
5137:
5096:
4867:
4863:
4859:
4857:
4794:
4732:
4646:
4552:
4459:If we write
4458:
4218:
4117:
4049:is given by
4046:
4042:
4040:
3754:as follows:
3751:
3749:
3739:
3738:
3518:
3514:
3508:
3442:
3316:
3255:
3155:
3150:
3146:
3142:
3138:
3134:
3130:
3126:
3122:
3118:
3114:
3110:
3106:
3102:
3098:
3094:
3090:
3086:
3082:
3078:
3074:
3070:
3066:
3062:
3058:
3054:
3050:
3048:
3043:
3041:
3002:
2961:
2884:
2882:
2795:
2707:
2705:
2628:
2501:
2497:
2479:
2477:
2375:
2293:
2146:
2080:
1925:
1866:
1856:
1850:
1840:
1830:
1820:
1794:
1790:
1747:
1737:
1733:
1729:
1721:
1717:
1712:(work), the
1709:
1703:
1690:
1680:
1652:
1648:
1642:
1438:Carathéodory
1369:Heat engines
1341:
1330:
1319:
1301:Motive power
1286:
1049:
946:Free entropy
917:
417:
416: /
406:
405: /
397:introduction
390:
389: /
328:
291:Heat engines
78: /
10449:von Neumann
10319:force field
10311:percolation
9936:free energy
9476:Hooke's law
1734:free energy
1672:temperature
1260:Synergetics
941:Free energy
387:Temperature
248:Quasistatic
243:Isenthalpic
200:Instruments
190:Equilibrium
142:Open system
76:Equilibrium
58:Statistical
10519:Categories
10306:Heisenberg
10064:2016-12-06
10027:2012-08-19
9974:References
8130:where the
3256:where the
1744:Definition
1695:explosives
1676:isothermal
1572:Nucleation
1416:Scientists
1220:Philosophy
933:Potentials
296:Heat pumps
253:Polytropic
238:Isentropic
228:Isothermal
10429:Boltzmann
10352:H-theorem
10230:Ensembles
10009:Gold Book
9897:−
9868:∑
9834:release.
9830:in their
9778:μ
9768:∑
9755:−
9743:ε
9730:σ
9687:μ
9677:∑
9664:−
9652:ε
9639:ε
9525:ε
9490:σ
9449:ε
9419:σ
9370:μ
9360:∑
9341:−
9329:ε
9310:σ
9297:∑
9216:~
9207:≤
9178:~
9126:
9120:−
9114:β
9111:−
9102:≥
9068:
9062:−
9056:β
9053:−
9045:
9029:
9002:
8954:≥
8925:
8909:
8848:≡
8777:
8771:≥
8746:
8703:~
8690:
8684:−
8672:~
8662:β
8659:−
8640:~
8626:
8591:~
8515:
8500:~
8484:∑
8480:≥
8464:~
8450:
8435:~
8419:∑
8355:~
8339:∑
8276:~
8240:~
8211:~
8180:~
8145:~
8110:~
8087:~
8077:β
8074:−
8066:
8038:~
8021:~
8012:β
8009:−
8001:
7968:~
7908:β
7905:−
7897:
7840:~
7763:~
7728:~
7719:≤
7690:~
7625:
7619:−
7613:β
7610:−
7602:≥
7587:~
7574:
7568:−
7562:~
7553:β
7550:−
7514:~
7486:~
7477:β
7474:−
7466:
7445:~
7386:β
7383:−
7375:
7324:~
7270:
7259:≥
7238:~
7224:
7165:−
7153:~
7132:∑
7128:≥
7100:~
7084:
7069:~
7053:∑
7015:−
7009:≥
6995:
6940:~
6924:
6909:~
6893:∑
6847:
6832:~
6816:∑
6812:≥
6796:~
6782:
6767:~
6751:∑
6714:~
6636:−
6627:⟩
6620:⟨
6617:≤
6603:and thus
6572:−
6563:⟩
6556:⟨
6544:⟩
6537:Δ
6534:⟨
6515:−
6506:⟩
6492:⟨
6483:~
6454:⟩
6448:Δ
6445:⟨
6395:~
6363:~
6354:≤
6322:⟩
6316:⟨
6303:⟩
6297:Δ
6294:⟨
6261:~
6225:~
6199:⟩
6193:⟨
6167:⟩
6161:⟨
6149:⟩
6142:⟨
6089:~
6026:⟩
6019:⟨
5987:⟩
5980:Δ
5977:⟨
5955:~
5890:Δ
5839:~
5784:~
5775:≤
5746:~
5714:⟩
5708:⟨
5695:~
5659:~
5569:∂
5561:∂
5510:∂
5502:∂
5492:−
5445:
5433:−
5372:∂
5364:∂
5347:μ
5313:∂
5305:∂
5295:−
5250:∂
5242:∂
5232:−
5186:μ
5170:−
5157:−
5117:−
5071:
5059:−
5042:and that
4965:β
4962:−
4948:Ω
4921:Ω
4894:Ω
4890:
4818:
4767:−
4705:−
4696:β
4687:
4672:β
4619:β
4606:β
4594:β
4585:−
4573:
4528:β
4525:−
4516:β
4504:β
4474:β
4431:β
4425:β
4415:−
4396:∂
4388:
4382:∂
4373:β
4363:β
4360:∂
4352:
4346:∂
4331:
4296:
4250:β
4195:∂
4187:
4181:∂
4173:β
4136:∑
4094:∂
4079:∂
4073:−
4020:β
4017:∂
4009:
4003:∂
3997:−
3973:β
3970:−
3956:∑
3949:β
3946:∂
3942:∂
3937:−
3910:β
3907:−
3896:β
3893:∂
3889:∂
3884:−
3872:∑
3837:β
3834:−
3817:∑
3784:∑
3777:⟩
3771:⟨
3768:≡
3709:β
3706:−
3692:∑
3595:β
3552:β
3549:−
3443:Here the
3405:μ
3395:∑
3358:∑
3354:−
3341:−
3292:μ
3218:μ
3208:∑
3191:−
3178:−
3021:≤
3015:Δ
2982:Δ
2979:−
2976:≤
2929:Δ
2923:−
2914:Δ
2898:Δ
2850:Δ
2844:−
2838:Δ
2832:−
2823:Δ
2807:Δ
2763:Δ
2757:−
2721:Δ
2676:Δ
2670:−
2654:Δ
2599:Δ
2583:Δ
2537:Δ
2529:increase
2510:Δ
2448:−
2433:−
2393:−
2346:−
2331:−
2316:−
2264:−
2249:−
2117:−
2055:−
2046:δ
2008:δ
1977:δ
1954:δ
1905:δ
1893:δ
1799:chemistry
1765:−
1759:≡
1687:chemistry
1553:Waterston
1503:von Mayer
1458:de Donder
1448:Clapeyron
1428:Boltzmann
1423:Bernoulli
1384:Education
1355:Timelines
1139:−
1084:−
872:Equations
839:∂
792:∂
743:α
707:∂
660:∂
614:−
608:β
572:∂
525:∂
233:Adiabatic
223:Isochoric
209:Processes
170:Ideal gas
53:Classical
10439:Tsallis
10135:Atkins'
10085:(1986).
9946:Enthalpy
9922:See also
9832:IAPWS-95
9141:⟩
9107:⟨
9088:⟩
9038:⟨
8965:⟩
8959:⟨
8940:⟩
8918:⟨
8893:See here
8874:⟩
8852:⟨
8838:⟩
8832:⟨
8792:⟩
8786:⟨
8761:⟩
8739:⟨
8576:and the
8390:⟩
8368:⟨
8331:⟩
8325:⟨
8296:⟩
8290:⟨
8282:⟩
8267:⟨
8053:⟩
7984:⟨
7933:⟩
7880:⟨
7793:⟩
7640:⟩
7606:⟨
7598:⟩
7546:⟨
7534:we get:
7292:⟩
7263:⟨
7255:⟩
7217:⟨
6308:⟩
6276:⟨
6267:⟩
6252:⟨
6075:. Since
5701:⟩
5686:⟨
4989:, where
4912:, where
3319:relation
2488:relation
2478:Because
1691:pressure
1605:Category
1543:Thompson
1453:Clausius
1433:Bridgman
1287:Vis viva
1269:Theories
1203:Gas laws
995:Enthalpy
403:Pressure
218:Isobaric
175:Real gas
63:Chemical
46:Branches
10434:Shannon
10421:Entropy
10121:: 3–10.
9957:thermal
9472:elastic
4553:we get
2527:entropy
2486:, this
2000:yields
1845:entropy
1843:is the
1835:kelvins
1823:is the
1726:physics
1655:) is a
1528:Smeaton
1523:Rankine
1513:Onsager
1498:Maxwell
1493:Massieu
1198:Entropy
1193:General
1184:History
1174:Culture
1171:History
395: (
392:Entropy
329:italics
130:Systems
10283:Models
10191:Theory
10093:
9410:where
9269:stress
8313:where
6981:Since
6185:where
5911:where
5805:where
4858:where
3584:where
3133:, and
2525:, the
1996:for a
1926:where
1899:
1807:joules
1786:where
1710:Arbeit
1665:closed
1647:, the
1518:Planck
1508:Nernst
1483:Kelvin
1443:Carnot
733:
598:
466:
408:Volume
323:Note:
282:Cycles
111:Second
101:Zeroth
10492:chaos
10444:Rényi
10301:Potts
10296:Ising
10115:(PDF)
10014:IUPAC
9828:IAPWS
9824:water
6437:plus
1724:. In
1566:Other
1533:Stahl
1488:Lewis
1478:Joule
1468:Gibbs
1463:Duhem
156:State
116:Third
106:First
10374:and
10091:ISBN
9959:and
9930:and
7950:and
7826:and
7676:and
7427:and
6699:and
3141:and
3109:and
3073:and
2906:bath
2815:bath
2744:bath
2729:bath
2662:bath
2646:bath
2591:bath
2294:and
1867:The
1815:ergs
1661:work
1651:(or
1538:Tait
368:Heat
363:Work
93:Laws
10018:doi
9123:log
9065:log
9042:exp
9026:log
8999:log
8922:exp
8906:log
8774:exp
8743:exp
8687:log
8623:log
8512:log
8447:log
8063:exp
7998:exp
7894:exp
7778:by
7622:log
7571:log
7463:exp
7372:exp
7267:log
7221:log
7081:log
6992:log
6921:log
6844:log
6779:log
5442:log
5068:log
4887:log
4815:log
4684:log
4570:log
4486:as
4385:log
4349:log
4328:log
4293:log
4184:log
4006:log
3089:= −
3053:= −
1811:CGS
1801:) (
1736:or
1643:In
1381:Art
327:in
10521::
10247::
10117:.
10081:;
10057:.
10016:.
10012:.
9991:.
4311::
3501:.
3129:,
3121:,
3097:−
3061:−
3044:PV
2614:0.
2081:PV
1817:),
1813::
1809:,
1805::
1803:SI
1740:.
10266:G
10261:F
10256:H
10251:U
10176:e
10169:t
10162:v
10099:.
10067:.
10030:.
10020::
9995:.
9963:.
9903:,
9900:H
9892:i
9888:E
9882:i
9878:p
9872:i
9864:=
9861:F
9797:.
9792:i
9788:N
9782:i
9772:i
9764:+
9761:T
9758:S
9750:j
9747:i
9737:j
9734:i
9726:V
9721:2
9718:1
9713:=
9701:i
9697:N
9691:i
9681:i
9673:+
9670:T
9667:S
9659:l
9656:k
9646:j
9643:i
9633:l
9630:k
9627:j
9624:i
9620:C
9616:V
9611:2
9608:1
9603:=
9596:F
9569:F
9565:d
9537:,
9532:l
9529:k
9519:l
9516:k
9513:j
9510:i
9506:C
9502:=
9497:j
9494:i
9456:j
9453:i
9426:j
9423:i
9395:,
9390:i
9386:N
9381:d
9374:i
9364:i
9356:+
9353:T
9349:d
9344:S
9336:j
9333:i
9324:d
9317:j
9314:i
9304:j
9301:i
9293:V
9290:=
9287:F
9283:d
9255:V
9251:d
9247:p
9213:F
9204:F
9175:H
9146:r
9136:)
9133:Z
9130:(
9117:H
9098:]
9093:r
9083:)
9078:)
9075:Z
9072:(
9059:H
9049:(
9033:[
9023:=
9019:)
9014:r
9010:P
9006:(
8970:r
8962:X
8950:]
8945:r
8935:)
8932:X
8929:(
8913:[
8870:r
8866:|
8863:Y
8860:|
8856:r
8843:r
8835:Y
8802:)
8797:r
8789:X
8781:(
8766:r
8756:)
8753:X
8750:(
8709:)
8700:Z
8694:(
8679:r
8669:E
8656:=
8652:)
8647:r
8637:P
8630:(
8598:r
8588:P
8562:r
8558:P
8532:)
8527:r
8523:P
8519:(
8507:r
8497:P
8488:r
8476:)
8471:r
8461:P
8454:(
8442:r
8432:P
8423:r
8386:r
8382:|
8379:H
8376:|
8372:r
8362:r
8352:P
8343:r
8335:=
8328:H
8293:H
8286:=
8273:H
8237:H
8208:H
8177:H
8152:r
8142:E
8107:Z
8100:)
8094:r
8084:E
8070:(
8057:=
8049:r
8045:|
8035:Z
8028:]
8018:H
8005:[
7992:|
7988:r
7980:=
7975:r
7965:P
7929:r
7925:|
7920:Z
7915:]
7911:H
7901:[
7888:|
7884:r
7876:=
7871:r
7867:P
7837:H
7814:H
7790:r
7787:|
7760:H
7725:F
7716:F
7687:H
7664:H
7635:)
7632:Z
7629:(
7616:H
7593:)
7584:Z
7578:(
7559:H
7511:Z
7504:]
7499:)
7496:r
7493:(
7483:H
7470:[
7457:=
7452:r
7442:P
7409:Z
7404:]
7399:)
7396:r
7393:(
7389:H
7379:[
7366:=
7361:r
7357:P
7331:r
7321:P
7287:)
7282:r
7278:P
7274:(
7250:)
7245:r
7235:P
7228:(
7186:0
7183:=
7179:)
7173:r
7169:P
7160:r
7150:P
7142:(
7136:r
7124:)
7117:r
7113:P
7107:r
7097:P
7088:(
7076:r
7066:P
7057:r
7023:x
7020:1
7012:1
7005:)
7002:x
6999:(
6964:)
6957:r
6953:P
6947:r
6937:P
6928:(
6916:r
6906:P
6897:r
6864:)
6859:r
6855:P
6851:(
6839:r
6829:P
6820:r
6808:)
6803:r
6793:P
6786:(
6774:r
6764:P
6755:r
6721:r
6711:P
6685:r
6681:P
6652:.
6647:0
6643:S
6639:T
6631:0
6623:H
6614:F
6588:,
6583:0
6579:S
6575:T
6567:0
6559:H
6553:=
6548:0
6540:H
6531:+
6526:0
6522:S
6518:T
6510:0
6500:0
6496:H
6489:=
6480:F
6451:H
6423:0
6419:H
6392:F
6360:F
6351:F
6325:,
6319:H
6313:=
6300:H
6291:+
6286:0
6282:H
6271:=
6258:H
6222:H
6196:X
6170:.
6164:X
6158:=
6153:0
6145:X
6117:0
6113:H
6086:H
6061:0
6057:H
6046:X
6030:0
6022:X
5996:.
5991:0
5983:H
5974:+
5969:0
5965:H
5961:=
5952:H
5924:0
5920:H
5896:,
5893:H
5887:+
5882:0
5878:H
5874:=
5871:H
5836:F
5813:F
5790:,
5781:F
5772:F
5743:H
5717:,
5711:H
5705:=
5692:H
5656:H
5633:H
5598:.
5593:T
5590:,
5587:N
5582:|
5578:)
5572:Q
5564:F
5555:(
5547:=
5544:V
5540:,
5535:N
5532:,
5529:T
5524:|
5519:)
5513:B
5505:F
5496:(
5484:=
5481:m
5451:,
5448:Z
5439:T
5436:k
5430:=
5427:F
5401:.
5396:V
5393:,
5390:T
5385:|
5381:)
5375:N
5367:F
5358:(
5350:=
5343:,
5338:N
5335:,
5332:T
5327:|
5322:)
5316:V
5308:F
5299:(
5287:=
5284:P
5280:,
5275:N
5272:,
5269:V
5264:|
5259:)
5253:T
5245:F
5236:(
5224:=
5221:S
5196:,
5193:N
5190:d
5183:+
5180:V
5177:d
5173:P
5167:T
5164:d
5160:S
5154:=
5151:F
5148:d
5123:S
5120:T
5114:U
5111:=
5108:F
5077:.
5074:Z
5065:T
5062:k
5056:=
5053:F
5030:0
5027:=
5024:c
5002:0
4998:U
4973:0
4969:U
4958:e
4952:0
4925:0
4898:0
4884:k
4881:=
4878:S
4868:T
4864:c
4860:c
4843:,
4840:c
4837:+
4832:T
4829:U
4824:+
4821:Z
4812:k
4809:=
4806:S
4780:.
4777:x
4774:d
4770:X
4764:S
4761:d
4757:T
4754:=
4751:U
4748:d
4718:.
4715:x
4712:d
4708:X
4702:)
4699:U
4693:+
4690:Z
4681:(
4678:d
4669:1
4664:=
4661:U
4658:d
4632:.
4629:x
4626:d
4622:X
4616:+
4613:U
4610:d
4603:+
4600:)
4597:U
4591:(
4588:d
4582:=
4579:)
4576:Z
4567:(
4564:d
4538:,
4535:U
4532:d
4522:)
4519:U
4513:(
4510:d
4507:=
4501:d
4497:U
4471:d
4467:U
4444:.
4441:x
4438:d
4434:X
4428:+
4422:d
4418:U
4412:=
4409:x
4406:d
4399:x
4391:Z
4376:+
4370:d
4355:Z
4340:=
4337:)
4334:Z
4325:(
4322:d
4299:Z
4273:x
4270:d
4247:d
4227:x
4204:.
4198:x
4190:Z
4170:1
4165:=
4160:r
4156:X
4150:r
4146:P
4140:r
4132:=
4129:X
4103:.
4097:x
4087:r
4083:E
4070:=
4065:r
4061:X
4047:x
4043:r
4026:.
4012:Z
3994:=
3989:Z
3981:r
3977:E
3966:e
3960:r
3931:=
3926:Z
3918:r
3914:E
3903:e
3876:r
3868:=
3863:Z
3857:r
3853:E
3845:r
3841:E
3830:e
3821:r
3813:=
3808:r
3804:E
3798:r
3794:P
3788:r
3780:=
3774:E
3765:U
3752:Z
3740:Z
3724:.
3717:i
3713:E
3702:e
3696:i
3688:=
3685:Z
3664:r
3642:r
3638:E
3616:,
3610:T
3607:k
3603:1
3598:=
3572:,
3567:Z
3560:r
3556:E
3545:e
3539:=
3534:r
3530:P
3519:i
3515:r
3483:i
3479:X
3456:i
3452:x
3428:.
3423:j
3419:N
3415:d
3409:j
3399:j
3391:+
3386:i
3382:x
3378:d
3372:i
3368:X
3362:i
3351:T
3348:d
3344:S
3338:=
3335:F
3332:d
3296:j
3269:j
3265:N
3241:,
3236:j
3232:N
3228:d
3222:j
3212:j
3204:+
3201:V
3198:d
3194:P
3188:T
3185:d
3181:S
3175:=
3172:F
3169:d
3156:j
3151:j
3147:N
3143:V
3139:T
3135:P
3131:V
3127:T
3123:V
3119:T
3117:(
3115:P
3111:V
3107:T
3103:V
3101:d
3099:P
3095:T
3093:d
3091:S
3087:F
3083:F
3079:F
3075:V
3071:T
3067:V
3065:d
3063:P
3059:T
3057:d
3055:S
3051:F
3027:,
3024:0
3018:F
2988:.
2985:F
2973:W
2947:.
2942:T
2938:W
2935:+
2932:F
2920:=
2917:S
2911:+
2902:S
2885:T
2868:.
2863:T
2859:W
2856:+
2853:S
2847:T
2841:U
2829:=
2826:S
2820:+
2811:S
2781:.
2776:T
2772:W
2769:+
2766:U
2754:=
2749:T
2740:Q
2734:=
2725:S
2708:T
2691:.
2688:)
2685:W
2682:+
2679:U
2673:(
2667:=
2658:U
2651:=
2642:Q
2611:=
2608:W
2605:+
2602:U
2596:+
2587:U
2560:W
2540:S
2513:U
2480:F
2463:.
2460:V
2456:d
2451:p
2445:T
2441:d
2436:S
2430:=
2427:F
2423:d
2399:S
2396:T
2390:U
2387:=
2384:F
2361:.
2358:V
2354:d
2349:p
2343:T
2339:d
2334:S
2328:=
2325:)
2322:S
2319:T
2313:U
2310:(
2306:d
2279:,
2276:V
2272:d
2267:p
2261:T
2257:d
2252:S
2246:)
2243:S
2240:T
2237:(
2233:d
2229:=
2226:U
2222:d
2198:T
2194:d
2190:S
2187:+
2183:S
2179:d
2175:T
2172:=
2169:)
2166:S
2163:T
2160:(
2156:d
2132:.
2129:V
2125:d
2120:p
2114:S
2110:d
2105:T
2102:=
2099:U
2095:d
2067:V
2063:d
2058:p
2052:=
2049:W
2026:S
2022:d
2017:T
2014:=
2011:Q
1980:W
1957:Q
1934:U
1911:,
1908:W
1902:+
1896:Q
1890:=
1887:U
1883:d
1857:U
1841:S
1831:T
1821:U
1795:A
1791:F
1774:,
1771:S
1768:T
1762:U
1756:F
1730:F
1718:A
1674:(
1632:e
1625:t
1618:v
1145:S
1142:T
1136:H
1133:=
1130:)
1127:p
1124:,
1121:T
1118:(
1115:G
1090:S
1087:T
1081:U
1078:=
1075:)
1072:V
1069:,
1066:T
1063:(
1060:A
1035:V
1032:p
1029:+
1026:U
1023:=
1020:)
1017:p
1014:,
1011:S
1008:(
1005:H
980:)
977:V
974:,
971:S
968:(
965:U
842:T
818:V
795:V
771:1
746:=
710:p
686:V
663:V
639:1
611:=
575:T
551:N
528:S
504:T
479:=
476:c
399:)
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