1920:
example of this is a gas of identical particles whose state is written in terms of the particles' individual positions and momenta: when two particles are exchanged, the resulting point in phase space is different, and yet it corresponds to an identical physical state of the system. It is important in statistical mechanics (a theory about physical states) to recognize that the phase space is just a mathematical construction, and to not naively overcount actual physical states when integrating over phase space. Overcounting can cause serious problems:
3777:
466:
49:
2717:
517:) âa statistical ensemble where the total energy of the system and the number of particles in the system are each fixed to particular values; each of the members of the ensemble are required to have the same total energy and particle number. The system must remain totally isolated (unable to exchange energy or particles with its environment) in order to stay in statistical equilibrium.
2499:
3432:
Technically, there are some phases where the permutation of particles does not even yield a distinct specific phase: for example, two similar particles can share the exact same trajectory, internal state, etc.. However, in classical mechanics these phases only make up an infinitesimal fraction of the
498:"We may imagine a great number of systems of the same nature, but differing in the configurations and velocities which they have at a given instant, and differing in not merely infinitesimally, but it may be so as to embrace every conceivable combination of configuration and velocities..." J. W. Gibbs
492:
The study of thermodynamics is concerned with systems that appear to human perception to be "static" (despite the motion of their internal parts), and which can be described simply by a set of macroscopically observable variables. These systems can be described by statistical ensembles that depend on
415:
The concept of an equilibrium or stationary ensemble is crucial to many applications of statistical ensembles. Although a mechanical system certainly evolves over time, the ensemble does not necessarily have to evolve. In fact, the ensemble will not evolve if it contains all past and future phases of
3024:
for a system in a physics lab: For example, the procedure might involve a physical apparatus and some protocols for manipulating the apparatus. As a result of this preparation procedure, some system is produced and maintained in isolation for some small period of time. By repeating this laboratory
1919:
Typically, the phase space contains duplicates of the same physical state in multiple distinct locations. This is a consequence of the way that a physical state is encoded into mathematical coordinates; the simplest choice of coordinate system often allows a state to be encoded in multiple ways. An
1991:
As mentioned above, the classic example of this overcounting is for a fluid system containing various kinds of particles, where any two particles of the same kind are indistinguishable and exchangeable. When the state is written in terms of the particles' individual positions and momenta, then the
1792:
over microstates, it is necessary to somehow partition the phase space into blocks that are distributed representing the different states of the system in a fair way. It turns out that the correct way to do this simply results in equal-sized blocks of canonical phase space, and so a microstate in
1981:
introduced above, which is a whole number that represents how many ways a physical state can be represented in phase space. Its value does not vary with the continuous canonical coordinates, so overcounting can be corrected simply by integrating over the full range of canonical coordinates, then
559:
The calculations that can be made using each of these ensembles are explored further in their respective articles. Other thermodynamic ensembles can be also defined, corresponding to different physical requirements, for which analogous formulae can often similarly be derived. For example, in the
554:
are specified. The grand canonical ensemble is appropriate for describing an open system: one which is in, or has been in, weak contact with a reservoir (thermal contact, chemical contact, radiative contact, electrical contact, etc.). The ensemble remains in statistical equilibrium if the system
1397:
If the number of parts in the system is allowed to vary among the systems in the ensemble (as in a grand ensemble where the number of particles is a random quantity), then it is a probability distribution over an extended phase space that includes further variables such as particle numbers
1649:
368:, considered all at once, each of which represents a possible state that the real system might be in. In other words, a statistical ensemble is a set of systems of particles used in statistical mechanics to describe a single system. The concept of an ensemble was introduced by
2216:
1950:
A crude way to remove the overcounting would be to manually define a subregion of phase space that includes each physical state only once and then exclude all other parts of phase space. In a gas, for example, one could include only those phases where the particles'
726:. The density matrix provides a fully general tool that can incorporate both quantum uncertainties (present even if the state of the system were completely known) and classical uncertainties (due to a lack of knowledge) in a unified manner. Any physical observable
1779:
3221:
537:
with a heat bath. In order to be in statistical equilibrium, the system must remain totally closed (unable to exchange particles with its environment) and may come into weak thermal contact with other systems that are described by ensembles with the same
407:
properties. For many important physical cases, it is possible to calculate averages directly over the whole of the thermodynamic ensemble, to obtain explicit formulas for many of the thermodynamic quantities of interest, often in terms of the appropriate
1946:
It is in general difficult to find a coordinate system that uniquely encodes each physical state. As a result, it is usually necessary to use a coordinate system with multiple copies of each state, and then to recognize and remove the overcounting.
2905:
391:
The ensemble formalises the notion that an experimenter repeating an experiment again and again under the same macroscopic conditions, but unable to control the microscopic details, may expect to observe a range of different outcomes.
1228:
3354:, i. e., the principle of conservation of extension in canonical phase space for Hamiltonian mechanics. This can also be demonstrated starting with the conception of an ensemble as a multitude of systems. See Gibbs'
3597:
Heath Turner, C.; Brennan, John K.; LĂsal, Martin; Smith, William R.; Karl
Johnson, J.; Gubbins, Keith E. (2008). "Simulation of chemical reaction equilibria by the reaction ensemble Monte Carlo method: a review".
843:
2990:
In the discussion given so far, while rigorous, we have taken for granted that the notion of an ensemble is valid a priori, as is commonly done in physical context. What has not been shown is that the ensemble
1957:
coordinates are sorted in ascending order. While this would solve the problem, the resulting integral over phase space would be tedious to perform due to its unusual boundary shape. (In this case, the factor
1518:
1974:
A simpler way to correct the overcounting is to integrate over all of phase space but to reduce the weight of each phase in order to exactly compensate the overcounting. This is accomplished by the factor
493:
a few observable parameters, and which are in statistical equilibrium. Gibbs noted that different macroscopic constraints lead to different types of ensembles, with particular statistical characteristics.
588:
The precise mathematical expression for a statistical ensemble has a distinct form depending on the type of mechanics under consideration (quantum or classical). In the classical case, the ensemble is a
2494:{\displaystyle {\bar {A}}={\frac {\int {Ae^{-\beta H(q_{1},q_{2},\dots ,q_{M},p_{1},p_{2},\dots ,p_{N})}\,d\tau }}{\int {e^{-\beta H(q_{1},q_{2},\dots ,q_{M},p_{1},p_{2},\dots ,p_{N})}\,d\tau }}},}
3292:
1001:
1259:
1660:
1924:
Dependence of derived quantities (such as entropy and chemical potential) on the choice of coordinate system, since one coordinate system might show more or less overcounting than another.
2061:
1857:
3119:
1793:
classical mechanics is an extended region in the phase space of canonical coordinates that has a particular volume. In particular, the probability density function in phase space,
1060:
960:
1506:
can be written as a function of the system's phase. The expectation value of any such quantity is given by an integral over the entire phase space of this quantity weighted by
2592:
915:
724:
3234:
questions to the lattice of closed subspaces of a
Hilbert space. With some additional technical assumptions one can then infer that states are given by density operators
2534:
1126:
1097:
875:
757:
17:
2791:
2685:
2560:
2660:
2633:
777:
379:
is a specific variety of statistical ensemble that, among other properties, is in statistical equilibrium (defined below), and is used to derive the properties of
4098:
3738:
605:; the microstates are the result of partitioning phase space into equal-sized units, although the size of these units can be chosen somewhat arbitrarily.
1784:
Phase space is a continuous space containing an infinite number of distinct physical states within any small region. In order to connect the probability
3409:âeach physical orientation can be encoded in two ways. If this encoding is used without correcting the overcounting, then the entropy will be higher by
3374:, leading to unit-dependence in the values of some thermodynamic quantities like entropy and chemical potential. Since the advent of quantum mechanics,
1142:
2084:
The formulation of statistical ensembles used in physics has now been widely adopted in other fields, in part because it has been recognized that the
2738:
2123:
by means of nearest-neighbor interactions between spins. The statistical formulation of the principle of locality is now seen to be a form of the
3297:
We see this reflects the definition of quantum states in general: A quantum state is a mapping from the observables to their expectation values.
335:
3507:
1099:(Hamiltonian). The grand canonical ensemble is additionally a function of the particle number, measured by the total particle number operator
789:
3793:
2171:
2164:
1988:
does vary strongly with discrete variables such as numbers of particles, and so it must be applied before summing over particle numbers.
129:
1644:{\displaystyle \langle X\rangle =\sum _{N_{1}=0}^{\infty }\ldots \sum _{N_{s}=0}^{\infty }\int \ldots \int \rho X\,dp_{1}\ldots dq_{n}.}
3798:
3731:
2696:
1905:
influences the offsets of quantities such as entropy and chemical potential, and so it is important to be consistent with the value of
3052:
sequence of systems. The systems are similar in that they were all produced in the same way. This infinite sequence is an ensemble.
3063:. Again, the testing procedure involves a physical apparatus and some protocols; as a result of the testing procedure we obtain a
1250:, are the elements of a complete and orthogonal basis. (Note that in other bases, the density matrix is not necessarily diagonal.)
529:)âa statistical ensemble where the energy is not known exactly but the number of particles is fixed. In place of the energy, the
580:, deviations to this rule occurs under conditions that state-variables are non-convex, such as small molecular measurements.
3724:
598:
555:
comes into weak contact with other systems that are described by ensembles with the same temperature and chemical potential.
3351:
3321:
1287:
3244:
965:
328:
3467:
2160:
1774:{\displaystyle \sum _{N_{1}=0}^{\infty }\ldots \sum _{N_{s}=0}^{\infty }\int \ldots \int \rho \,dp_{1}\ldots dq_{n}=1.}
1434:
1353:
400:
163:
2781:, for a quantum system in thermal equilibrium with its environment, the weighted average takes the form of a sum over
1889:
is an overcounting correction factor (see below), generally dependent on the number of particles and similar concerns.
3566:
3538:
2764:
550:)âa statistical ensemble where neither the energy nor particle number are fixed. In their place, the temperature and
533:
is specified. The canonical ensemble is appropriate for describing a closed system which is in, or has been in, weak
2746:
3326:
2916:
409:
171:
91:
3216:{\displaystyle \sigma (E)=\lim _{N\rightarrow \infty }{\frac {1}{N}}\sum _{k=1}^{N}\operatorname {Meas} (E,X_{k})}
1998:
1811:
4093:
3992:
2742:
1278:
In classical mechanics, an ensemble is represented by a probability density function defined over the system's
486:
321:
4088:
2778:
396:
3761:
2093:
2075:
482:
96:
1016:
924:
693:
A statistical ensemble in quantum mechanics (also known as a mixed state) is most often represented by a
86:
81:
3869:
3785:
3512:
3311:
2187:
101:
2565:
364:) is an idealization consisting of a large number of virtual copies (sometimes infinitely many) of a
31:
1899:
can be chosen arbitrarily, the notional size of a microstate is also arbitrary. Still, the value of
884:
700:
4045:
4028:
3711:
3634:
3331:
2967:
2727:
2183:
2111:: that all interactions are only between neighboring atoms or nearby molecules. Thus, for example,
1939:
590:
542:
478:
442:
155:
3680:
3892:
3882:
3808:
2935:
2731:
2131:. Thus, the general notion of a statistical ensemble with nearest-neighbor interactions leads to
2112:
1303:
1283:
1063:
577:
509:
470:
217:
139:
63:
3423:. This does not actually lead to any observable error since it only causes unobservable offsets.
2510:
1102:
1073:
851:
733:
3939:
2900:{\displaystyle {\bar {A}}={\frac {\sum _{i}A_{i}e^{-\beta E_{i}}}{\sum _{i}e^{-\beta E_{i}}}}.}
2120:
780:
614:
1132:
in the orthogonal basis of states that simultaneously diagonalize each conserved variable. In
3824:
3747:
3434:
3406:
2924:
2605:
2148:
2108:
1294:
1263:
353:
234:
40:
2667:
2545:
601:. In classical mechanics, the ensemble is instead written as a probability distribution in
3967:
3502:
3395:
choice of coordinate system used for representing orientations of three-dimensional objects
2638:
2611:
1433:
is how many different kinds of particles there are). The ensemble is then represented by a
1133:
1070:
are strictly functions of the total energy, which is measured by the total energy operator
1007:
762:
560:
reaction ensemble, particle number fluctuations are only allowed to occur according to the
380:
277:
8:
4033:
4012:
2995:(not the consequent results) is a precisely defined object mathematically. For instance,
2199:
2179:
2132:
2092:
serves to maximize the entropy of a system, subject to a set of constraints: this is the
2079:
434:
272:
262:
74:
448:
The term "ensemble" is often used in physics and the physics-influenced literature. In
4060:
3972:
3949:
3930:
3910:
3874:
3646:
2951:
2920:
2782:
2595:
2085:
1935:
1331:
1274:(red curve, lower figure). The initially compact ensemble becomes swirled up over time.
1067:
551:
521:
474:
449:
416:
the system. Such a statistical ensemble, one that does not change over time, is called
147:
3055:
In a laboratory setting, each one of these prepped systems might be used as input for
2919:
provides the complete framework for working with ensemble averages in thermodynamics,
4065:
3962:
3935:
3846:
3836:
3829:
3803:
3615:
3562:
3534:
3463:
2928:
2178:
obtained for a given physical quantity does not depend on the ensemble chosen at the
2163:
of a system, according to the distribution of the system on its micro-states in this
676:
613:
Putting aside for the moment the question of how statistical ensembles are generated
565:
453:
297:
239:
3957:
3656:
3607:
2174:
chosen, its mathematical expression varies from ensemble to ensemble. However, the
2136:
594:
292:
267:
116:
4007:
1992:
overcounting related to the exchange of identical particles is corrected by using
1971:, and the integral would be restricted to the selected subregion of phase space.)
1223:{\displaystyle {\hat {\rho }}=\sum _{i}P_{i}|\psi _{i}\rangle \langle \psi _{i}|,}
4002:
3997:
3925:
3920:
3887:
3814:
3394:
3381:
2124:
1129:
576:
In thermodynamic limit all ensembles should produce identical observables due to
534:
282:
244:
224:
1927:
Erroneous conclusions that are inconsistent with physical experience, as in the
4050:
3766:
3660:
3526:
3306:
2688:
2128:
1928:
1271:
1062:) can be written solely as a function of conserved variables. For example, the
1003:(this essentially is the condition that the probabilities must add up to one).
694:
688:
287:
194:
106:
58:
3611:
4082:
4038:
3619:
3227:
2089:
1258:
561:
369:
597:, is a way of assigning a probability distribution over the results of each
395:
The notional size of ensembles in thermodynamics, statistical mechanics and
4055:
3004:
1875:, setting the extent of the microstate and providing correct dimensions to
307:
3437:
zero) and so they do not contribute to any volume integral in phase space.
469:
Visual representation of five statistical ensembles (from left to right):
3864:
3859:
3384:
in order to obtain a semiclassical correspondence with quantum mechanics.
3316:
2207:
2116:
2097:
1279:
1270:(top). Each system consists of one massive particle in a one-dimensional
1267:
617:, we should be able to perform the following two operations on ensembles
602:
530:
438:
404:
302:
204:
199:
189:
3716:
465:
3854:
3554:
3402:
918:
838:{\displaystyle \langle X\rangle =\operatorname {Tr} ({\hat {X}}\rho ).}
3075:
applied to each prepared system, we obtain a sequence of values Meas (
48:
3915:
1286:, the density function (the ensemble) evolves over time according to
759:. The expectation value of this operator on the statistical ensemble
3226:
For quantum mechanical systems, an important assumption made in the
2716:
437:
from the full set of possible states. For example, a collection of
433:
The word "ensemble" is also used for a smaller set of possibilities
3819:
3398:
3049:
2107:
In addition, statistical ensembles in physics are often built on a
2101:
878:
229:
3651:
3393:
In some cases the overcounting error is benign. An example is the
3984:
593:
over the microstates. In quantum mechanics, this notion, due to
349:
3017:
In this section, we attempt to partially answer this question.
2958:) with its surroundings (usually a heat bath), but the volume (
1799:, is related to the probability distribution over microstates,
504:
Three important thermodynamic ensembles were defined by Gibbs:
365:
1654:
The condition of probability normalization applies, requiring
3596:
1871:
is an arbitrary but predetermined constant with the units of
679:
of statistical ensembles have the structure of a convex set.
652:, then produce a new ensemble by probabilistic sampling from
111:
2096:. This principle has now been widely applied to problems in
1006:
In general, the ensemble evolves over time according to the
3712:
Monte Carlo applet applied in statistical physics problems.
3533:. San Francisco: W.H. Freeman and Company. pp. 31 ff.
3367:(Historical note) Gibbs' original ensemble effectively set
2608:
of the classical system in terms of the set of coordinates
2175:
2156:
1013:
Equilibrium ensembles (those that do not evolve over time,
3110:). Each one of these values is a 0 (or no) or a 1 (yes).
3048:, which in our mathematical idealization, we assume is an
962:), etc. The density matrix must always have a trace of 1:
445:
iteration is called an ensemble in some of the literature.
3635:""Ensemble inequivalence in single-molecule experiments""
3416:
per rotatable object and the chemical potential lower by
1982:
dividing the result by the overcounting factor. However,
1914:
3688:
George Mason
University Physics and Astronomy Department
3230:
approach to quantum mechanics is the identification of
3013:
It is not clear how to physically generate an ensemble.
2135:, which again find broad applicability; for example in
3025:
preparation procedure we obtain a sequence of systems
2954:
represents a closed system which can exchange energy (
27:
Idealization of a large number of atomic-sized systems
3525:
3247:
3122:
2970:
represents an open system which can exchange energy (
2794:
2670:
2641:
2614:
2568:
2548:
2513:
2219:
2001:
1814:
1663:
1521:
1145:
1105:
1076:
1019:
968:
927:
887:
854:
792:
765:
736:
703:
403:
the system could be in, consistent with its observed
3553:
1297:
with a defined number of parts, the phase space has
730:
in quantum mechanics can be written as an operator,
3350:This equal-volume partitioning is a consequence of
3287:{\displaystyle \sigma (E)=\operatorname {Tr} (ES).}
2695:The denominator in this expression is known as the
996:{\displaystyle \operatorname {Tr} {\hat {\rho }}=1}
608:
3457:
3451:
3286:
3215:
2899:
2679:
2654:
2627:
2586:
2554:
2528:
2493:
2193:
2055:
1851:
1773:
1643:
1282:. While an individual system evolves according to
1222:
1120:
1091:
1054:
995:
954:
909:
869:
837:
771:
751:
718:
383:from the laws of classical or quantum mechanics.
4080:
3139:
2706:
848:This can be used to evaluate averages (operator
2938:represents an isolated system in which energy (
2536:is the ensemble average of the system property
2170:Since the ensemble average is dependent on the
2066:This is known as "correct Boltzmann counting".
3508:Elementary Principles in Statistical Mechanics
2127:in the broad sense; nearest neighbors are now
4099:Philosophy of thermal and statistical physics
3732:
3632:
3586:. Green & Co, London, New York: Longmans.
2985:
2910:
329:
3681:"Statistical mechanics of classical systems"
2056:{\displaystyle C=N_{1}!N_{2}!\ldots N_{s}!.}
1528:
1522:
1199:
1196:
799:
793:
399:can be very large, including every possible
2745:. Unsourced material may be challenged and
1852:{\displaystyle \rho ={\frac {1}{h^{n}C}}P,}
1416:(second kind of particle), and so on up to
18:Statistical ensemble (mathematical physics)
3739:
3725:
3633:SĂŒzen, M; Sega, M; Holm, C (18 May 2009).
3113:Assume the following time average exists:
2069:
386:
336:
322:
47:
3746:
3650:
2978:) with its surroundings, but the volume (
2765:Learn how and when to remove this message
2691:of the classical phase space of interest.
2477:
2354:
1735:
1608:
2635:and their conjugate generalized momenta
2159:of a quantity that is a function of the
1352:. The ensemble is then represented by a
1257:
464:
3497:
3495:
3493:
3491:
3489:
3487:
3485:
3483:
3481:
3479:
2206:takes the form of an integral over the
1253:
14:
4081:
3458:Rennie, Richard; Jonathan Law (2019).
1915:Correcting overcounting in phase space
1497:varies with the numbers of particles.
3720:
3626:
3581:
3547:
3519:
3501:
2785:, rather than a continuous integral:
682:
675:Under certain conditions, therefore,
599:complete set of commuting observables
3476:
2743:adding citations to reliable sources
2710:
1934:Foundational issues in defining the
1055:{\displaystyle d{\hat {\rho }}/dt=0}
955:{\displaystyle {\hat {X}}{\hat {Y}}}
245:Grand potential / Landau free energy
2142:
1128:. Such equilibrium ensembles are a
24:
3606:(2). Informa UK Limited: 119â146.
3380:is often taken to be equal to the
3149:
3071:answer. Given a testing procedure
1911:when comparing different systems.
1718:
1687:
1588:
1557:
1435:joint probability density function
1354:joint probability density function
583:
568:which are present in the system.
25:
4110:
3705:
3561:. Pergamon Press. pp. 9 ff.
3322:Liouville's theorem (Hamiltonian)
1964:introduced above would be set to
3775:
2715:
1788:in phase space to a probability
609:Requirements for representations
3673:
3531:Thermal Physics, Second Edition
3426:
3387:
3008:of particles inside a container
2962:) and the number of particles (
2946:) and the number of particles (
2915:The generalized version of the
2587:{\displaystyle {\frac {1}{kT}}}
2194:Classical statistical mechanics
3590:
3575:
3361:
3344:
3278:
3269:
3257:
3251:
3210:
3191:
3146:
3132:
3126:
2801:
2520:
2472:
2382:
2349:
2259:
2226:
1213:
1182:
1152:
1112:
1083:
1029:
981:
946:
934:
910:{\displaystyle {\hat {X}}^{2}}
895:
861:
829:
820:
811:
743:
719:{\displaystyle {\hat {\rho }}}
710:
571:
487:isoenthalpic-isobaric ensemble
427:
13:
1:
3444:
3003:exists (for example, is it a
2779:quantum statistical mechanics
2707:Quantum statistical mechanics
2699:and is denoted by the letter
637:are statistically equivalent.
460:
397:quantum statistical mechanics
3762:Principle of maximum entropy
3460:Oxford Dictionary of Physics
3327:MaxwellâBoltzmann statistics
2094:principle of maximum entropy
2076:Principle of maximum entropy
1491:. The number of coordinates
1427:(the last kind of particle;
1262:Evolution of an ensemble of
483:isobaric-isothermal ensemble
7:
3584:The Collected Works, Vol. 2
3397:. A simple encoding is the
3300:
2999:It is not clear where this
644:is a real number such that
87:Indistinguishable particles
10:
4115:
3786:Statistical thermodynamics
3661:10.1103/PhysRevE.79.051118
3529:; Herbert Kroemer (1980).
3312:Ensemble (fluid mechanics)
2986:Operational interpretation
2911:Canonical ensemble average
2529:{\displaystyle {\bar {A}}}
2202:with its environment, the
2198:For a classical system in
2073:
1407:(first kind of particle),
1121:{\displaystyle {\hat {N}}}
1092:{\displaystyle {\hat {H}}}
870:{\displaystyle {\hat {X}}}
779:is given by the following
752:{\displaystyle {\hat {X}}}
686:
29:
4021:
3983:
3948:
3903:
3845:
3784:
3773:
3754:
3612:10.1080/08927020801986564
3557:; Lifshitz, E.M. (1980).
3001:very large set of systems
420:and can be said to be in
32:Ensemble (disambiguation)
4046:Condensed matter physics
4029:Statistical field theory
3337:
3332:Replication (statistics)
2968:grand canonical ensemble
2966:) are all constant. The
2950:) are all constant. The
2184:grand canonical ensemble
1940:grand canonical ensemble
1500:Any mechanical quantity
1136:, the density matrix is
591:probability distribution
543:Grand canonical ensemble
479:grand canonical ensemble
443:Markov chain Monte Carlo
3904:Mathematical approaches
3893:Lennard-Jones potential
3809:thermodynamic potential
3513:Charles Scribner's Sons
3433:phase space (they have
2936:microcanonical ensemble
2121:ferromagnetic materials
2070:Ensembles in statistics
1304:generalized coordinates
1064:microcanonical ensemble
510:Microcanonical ensemble
471:microcanonical ensemble
422:statistical equilibrium
387:Physical considerations
130:Thermodynamic ensembles
82:Spinâstatistics theorem
3940:conformal field theory
3288:
3217:
3184:
2901:
2681:
2680:{\displaystyle d\tau }
2656:
2629:
2588:
2556:
2555:{\displaystyle \beta }
2530:
2495:
2057:
1853:
1775:
1722:
1691:
1645:
1592:
1561:
1275:
1224:
1122:
1093:
1056:
997:
956:
911:
871:
839:
773:
753:
720:
502:
489:
377:thermodynamic ensemble
4094:Statistical ensembles
3855:Ferromagnetism models
3748:Statistical mechanics
3503:Gibbs, Josiah Willard
3356:Elementary Principles
3289:
3218:
3164:
3022:preparation procedure
2925:statistical mechanics
2902:
2783:quantum energy states
2682:
2657:
2655:{\displaystyle p_{i}}
2630:
2628:{\displaystyle q_{i}}
2589:
2557:
2531:
2496:
2149:statistical mechanics
2109:principle of locality
2058:
1854:
1776:
1695:
1664:
1646:
1565:
1534:
1261:
1225:
1123:
1094:
1057:
998:
957:
912:
872:
840:
774:
772:{\displaystyle \rho }
754:
721:
495:
468:
381:thermodynamic systems
354:statistical mechanics
235:Helmholtz free energy
164:Isoenthalpicâisobaric
41:Statistical mechanics
4089:Equations of physics
3600:Molecular Simulation
3582:Gibbs, J.W. (1928).
3245:
3120:
2982:) is kept constant.
2792:
2739:improve this section
2668:
2639:
2612:
2566:
2546:
2511:
2217:
2186:is an example of an
2133:Markov random fields
1999:
1812:
1661:
1519:
1288:Liouville's equation
1284:Hamilton's equations
1254:Classical mechanical
1143:
1103:
1074:
1017:
1008:von Neumann equation
966:
925:
885:
852:
790:
763:
734:
701:
625:of the same system:
362:statistical ensemble
30:For other uses, see
4034:elementary particle
3799:partition functions
3559:Statistical Physics
3462:. pp. 458 ff.
3352:Liouville's theorem
2200:thermal equilibrium
2180:thermodynamic limit
2080:Markov random field
677:equivalence classes
578:Legendre transforms
172:Isothermalâisobaric
75:Particle statistics
4061:information theory
3968:correlation length
3963:Critical exponents
3950:Critical phenomena
3931:stochastic process
3911:Boltzmann equation
3804:equations of state
3284:
3213:
3153:
3020:Suppose we have a
2952:canonical ensemble
2921:information theory
2917:partition function
2897:
2867:
2822:
2697:partition function
2677:
2652:
2625:
2596:thermodynamic beta
2584:
2552:
2526:
2491:
2155:is defined as the
2086:canonical ensemble
2053:
1936:chemical potential
1849:
1771:
1641:
1276:
1220:
1170:
1118:
1089:
1068:canonical ensemble
1052:
993:
952:
907:
867:
835:
769:
749:
716:
683:Quantum mechanical
566:chemical reactions
552:chemical potential
522:Canonical ensemble
490:
475:canonical ensemble
456:is more prevalent.
450:probability theory
410:partition function
112:Anyonic statistics
4076:
4075:
4066:Boltzmann machine
3936:mean-field theory
3837:Maxwell relations
3639:Physical Review E
3162:
3138:
3061:testing procedure
2974:) and particles (
2929:quantum mechanics
2892:
2858:
2813:
2804:
2775:
2774:
2767:
2582:
2523:
2486:
2229:
2137:Hopfield networks
1841:
1332:canonical momenta
1295:mechanical system
1161:
1155:
1115:
1086:
1032:
984:
949:
937:
898:
864:
823:
746:
713:
664:with probability
656:with probability
454:probability space
401:microscopic state
346:
345:
240:Gibbs free energy
92:MaxwellâBoltzmann
16:(Redirected from
4106:
3958:Phase transition
3779:
3778:
3741:
3734:
3727:
3718:
3717:
3699:
3698:
3696:
3694:
3685:
3677:
3671:
3670:
3668:
3667:
3654:
3630:
3624:
3623:
3594:
3588:
3587:
3579:
3573:
3572:
3551:
3545:
3544:
3523:
3517:
3516:
3499:
3474:
3473:
3455:
3438:
3430:
3424:
3422:
3415:
3391:
3385:
3379:
3373:
3365:
3359:
3348:
3293:
3291:
3290:
3285:
3222:
3220:
3219:
3214:
3209:
3208:
3183:
3178:
3163:
3155:
3152:
2906:
2904:
2903:
2898:
2893:
2891:
2890:
2889:
2888:
2887:
2866:
2856:
2855:
2854:
2853:
2852:
2832:
2831:
2821:
2811:
2806:
2805:
2797:
2770:
2763:
2759:
2756:
2750:
2719:
2711:
2686:
2684:
2683:
2678:
2661:
2659:
2658:
2653:
2651:
2650:
2634:
2632:
2631:
2626:
2624:
2623:
2593:
2591:
2590:
2585:
2583:
2581:
2570:
2561:
2559:
2558:
2553:
2535:
2533:
2532:
2527:
2525:
2524:
2516:
2500:
2498:
2497:
2492:
2487:
2485:
2484:
2476:
2475:
2471:
2470:
2452:
2451:
2439:
2438:
2426:
2425:
2407:
2406:
2394:
2393:
2362:
2361:
2353:
2352:
2348:
2347:
2329:
2328:
2316:
2315:
2303:
2302:
2284:
2283:
2271:
2270:
2236:
2231:
2230:
2222:
2204:ensemble average
2153:ensemble average
2143:Ensemble average
2104:, and the like.
2062:
2060:
2059:
2054:
2046:
2045:
2030:
2029:
2017:
2016:
1987:
1980:
1970:
1963:
1956:
1910:
1904:
1898:
1888:
1880:
1874:
1870:
1858:
1856:
1855:
1850:
1842:
1840:
1836:
1835:
1822:
1804:
1798:
1780:
1778:
1777:
1772:
1764:
1763:
1748:
1747:
1721:
1716:
1709:
1708:
1690:
1685:
1678:
1677:
1650:
1648:
1647:
1642:
1637:
1636:
1621:
1620:
1591:
1586:
1579:
1578:
1560:
1555:
1548:
1547:
1511:
1505:
1496:
1490:
1432:
1426:
1415:
1406:
1393:
1351:
1329:
1323:
1302:
1249:
1245:
1229:
1227:
1226:
1221:
1216:
1211:
1210:
1195:
1194:
1185:
1180:
1179:
1169:
1157:
1156:
1148:
1134:braâket notation
1127:
1125:
1124:
1119:
1117:
1116:
1108:
1098:
1096:
1095:
1090:
1088:
1087:
1079:
1061:
1059:
1058:
1053:
1039:
1034:
1033:
1025:
1002:
1000:
999:
994:
986:
985:
977:
961:
959:
958:
953:
951:
950:
942:
939:
938:
930:
921:(using operator
916:
914:
913:
908:
906:
905:
900:
899:
891:
881:(using operator
876:
874:
873:
868:
866:
865:
857:
844:
842:
841:
836:
825:
824:
816:
778:
776:
775:
770:
758:
756:
755:
750:
748:
747:
739:
729:
725:
723:
722:
717:
715:
714:
706:
670:
651:
370:J. Willard Gibbs
338:
331:
324:
117:Braid statistics
51:
37:
36:
21:
4114:
4113:
4109:
4108:
4107:
4105:
4104:
4103:
4079:
4078:
4077:
4072:
4017:
3979:
3944:
3926:BBGKY hierarchy
3921:Vlasov equation
3899:
3888:depletion force
3881:Particles with
3841:
3780:
3776:
3771:
3750:
3745:
3708:
3703:
3702:
3692:
3690:
3683:
3679:
3678:
3674:
3665:
3663:
3631:
3627:
3595:
3591:
3580:
3576:
3569:
3552:
3548:
3541:
3527:Kittel, Charles
3524:
3520:
3500:
3477:
3470:
3456:
3452:
3447:
3442:
3441:
3431:
3427:
3417:
3410:
3392:
3388:
3382:Planck constant
3375:
3368:
3366:
3362:
3349:
3345:
3340:
3303:
3246:
3243:
3242:
3204:
3200:
3179:
3168:
3154:
3142:
3121:
3118:
3117:
3109:
3096:
3085:
3047:
3038:
3031:
2988:
2913:
2883:
2879:
2872:
2868:
2862:
2857:
2848:
2844:
2837:
2833:
2827:
2823:
2817:
2812:
2810:
2796:
2795:
2793:
2790:
2789:
2771:
2760:
2754:
2751:
2736:
2720:
2709:
2669:
2666:
2665:
2646:
2642:
2640:
2637:
2636:
2619:
2615:
2613:
2610:
2609:
2574:
2569:
2567:
2564:
2563:
2547:
2544:
2543:
2515:
2514:
2512:
2509:
2508:
2466:
2462:
2447:
2443:
2434:
2430:
2421:
2417:
2402:
2398:
2389:
2385:
2372:
2368:
2367:
2363:
2343:
2339:
2324:
2320:
2311:
2307:
2298:
2294:
2279:
2275:
2266:
2262:
2249:
2245:
2241:
2237:
2235:
2221:
2220:
2218:
2215:
2214:
2210:of the system:
2196:
2145:
2129:Markov blankets
2125:Markov property
2082:
2074:Main articles:
2072:
2041:
2037:
2025:
2021:
2012:
2008:
2000:
1997:
1996:
1983:
1976:
1965:
1959:
1952:
1917:
1906:
1900:
1894:
1884:
1876:
1872:
1866:
1831:
1827:
1826:
1821:
1813:
1810:
1809:
1800:
1794:
1759:
1755:
1743:
1739:
1717:
1704:
1700:
1699:
1686:
1673:
1669:
1668:
1662:
1659:
1658:
1632:
1628:
1616:
1612:
1587:
1574:
1570:
1569:
1556:
1543:
1539:
1538:
1520:
1517:
1516:
1507:
1501:
1492:
1488:
1479:
1472:
1463:
1456:
1447:
1437:
1428:
1425:
1417:
1414:
1408:
1405:
1399:
1391:
1382:
1375:
1366:
1356:
1350:
1341:
1335:
1325:
1322:
1313:
1307:
1298:
1256:
1247:
1243:
1234:
1212:
1206:
1202:
1190:
1186:
1181:
1175:
1171:
1165:
1147:
1146:
1144:
1141:
1140:
1130:diagonal matrix
1107:
1106:
1104:
1101:
1100:
1078:
1077:
1075:
1072:
1071:
1035:
1024:
1023:
1018:
1015:
1014:
976:
975:
967:
964:
963:
941:
940:
929:
928:
926:
923:
922:
901:
890:
889:
888:
886:
883:
882:
856:
855:
853:
850:
849:
815:
814:
791:
788:
787:
764:
761:
760:
738:
737:
735:
732:
731:
727:
705:
704:
702:
699:
698:
691:
685:
665:
645:
611:
586:
584:Representations
574:
535:thermal contact
463:
430:
389:
352:, specifically
342:
313:
312:
258:
250:
249:
225:Internal energy
220:
210:
209:
185:
177:
176:
156:Grand canonical
132:
122:
121:
77:
35:
28:
23:
22:
15:
12:
11:
5:
4112:
4102:
4101:
4096:
4091:
4074:
4073:
4071:
4070:
4069:
4068:
4063:
4058:
4051:Complex system
4048:
4043:
4042:
4041:
4036:
4025:
4023:
4019:
4018:
4016:
4015:
4010:
4005:
4000:
3995:
3989:
3987:
3981:
3980:
3978:
3977:
3976:
3975:
3970:
3960:
3954:
3952:
3946:
3945:
3943:
3942:
3933:
3928:
3923:
3918:
3913:
3907:
3905:
3901:
3900:
3898:
3897:
3896:
3895:
3890:
3879:
3878:
3877:
3872:
3867:
3862:
3851:
3849:
3843:
3842:
3840:
3839:
3834:
3833:
3832:
3827:
3822:
3817:
3806:
3801:
3796:
3790:
3788:
3782:
3781:
3774:
3772:
3770:
3769:
3767:ergodic theory
3764:
3758:
3756:
3752:
3751:
3744:
3743:
3736:
3729:
3721:
3715:
3714:
3707:
3706:External links
3704:
3701:
3700:
3672:
3625:
3589:
3574:
3567:
3546:
3539:
3518:
3475:
3469:978-0198821472
3468:
3449:
3448:
3446:
3443:
3440:
3439:
3425:
3386:
3360:
3342:
3341:
3339:
3336:
3335:
3334:
3329:
3324:
3319:
3314:
3309:
3307:Density matrix
3302:
3299:
3295:
3294:
3283:
3280:
3277:
3274:
3271:
3268:
3265:
3262:
3259:
3256:
3253:
3250:
3224:
3223:
3212:
3207:
3203:
3199:
3196:
3193:
3190:
3187:
3182:
3177:
3174:
3171:
3167:
3161:
3158:
3151:
3148:
3145:
3141:
3137:
3134:
3131:
3128:
3125:
3105:
3097:), ..., Meas (
3094:
3083:
3043:
3036:
3029:
3015:
3014:
3011:
2987:
2984:
2912:
2909:
2908:
2907:
2896:
2886:
2882:
2878:
2875:
2871:
2865:
2861:
2851:
2847:
2843:
2840:
2836:
2830:
2826:
2820:
2816:
2809:
2803:
2800:
2773:
2772:
2723:
2721:
2714:
2708:
2705:
2693:
2692:
2689:volume element
2676:
2673:
2663:
2649:
2645:
2622:
2618:
2599:
2580:
2577:
2573:
2551:
2541:
2522:
2519:
2502:
2501:
2490:
2483:
2480:
2474:
2469:
2465:
2461:
2458:
2455:
2450:
2446:
2442:
2437:
2433:
2429:
2424:
2420:
2416:
2413:
2410:
2405:
2401:
2397:
2392:
2388:
2384:
2381:
2378:
2375:
2371:
2366:
2360:
2357:
2351:
2346:
2342:
2338:
2335:
2332:
2327:
2323:
2319:
2314:
2310:
2306:
2301:
2297:
2293:
2290:
2287:
2282:
2278:
2274:
2269:
2265:
2261:
2258:
2255:
2252:
2248:
2244:
2240:
2234:
2228:
2225:
2195:
2192:
2144:
2141:
2115:, such as the
2113:lattice models
2071:
2068:
2064:
2063:
2052:
2049:
2044:
2040:
2036:
2033:
2028:
2024:
2020:
2015:
2011:
2007:
2004:
1944:
1943:
1932:
1929:mixing paradox
1925:
1916:
1913:
1891:
1890:
1882:
1860:
1859:
1848:
1845:
1839:
1834:
1830:
1825:
1820:
1817:
1782:
1781:
1770:
1767:
1762:
1758:
1754:
1751:
1746:
1742:
1738:
1734:
1731:
1728:
1725:
1720:
1715:
1712:
1707:
1703:
1698:
1694:
1689:
1684:
1681:
1676:
1672:
1667:
1652:
1651:
1640:
1635:
1631:
1627:
1624:
1619:
1615:
1611:
1607:
1604:
1601:
1598:
1595:
1590:
1585:
1582:
1577:
1573:
1568:
1564:
1559:
1554:
1551:
1546:
1542:
1537:
1533:
1530:
1527:
1524:
1484:
1477:
1468:
1461:
1452:
1445:
1421:
1412:
1403:
1387:
1380:
1371:
1364:
1346:
1339:
1318:
1311:
1272:potential well
1255:
1252:
1239:
1231:
1230:
1219:
1215:
1209:
1205:
1201:
1198:
1193:
1189:
1184:
1178:
1174:
1168:
1164:
1160:
1154:
1151:
1114:
1111:
1085:
1082:
1051:
1048:
1045:
1042:
1038:
1031:
1028:
1022:
992:
989:
983:
980:
974:
971:
948:
945:
936:
933:
904:
897:
894:
863:
860:
846:
845:
834:
831:
828:
822:
819:
813:
810:
807:
804:
801:
798:
795:
768:
745:
742:
712:
709:
695:density matrix
689:Density matrix
687:Main article:
684:
681:
673:
672:
638:
610:
607:
585:
582:
573:
570:
557:
556:
539:
518:
462:
459:
458:
457:
446:
429:
426:
388:
385:
344:
343:
341:
340:
333:
326:
318:
315:
314:
311:
310:
305:
300:
295:
290:
285:
280:
275:
270:
265:
259:
256:
255:
252:
251:
248:
247:
242:
237:
232:
227:
221:
216:
215:
212:
211:
208:
207:
202:
197:
192:
186:
183:
182:
179:
178:
175:
174:
166:
158:
150:
142:
140:Microcanonical
133:
128:
127:
124:
123:
120:
119:
114:
109:
107:Parastatistics
104:
99:
94:
89:
84:
78:
73:
72:
69:
68:
67:
66:
64:Kinetic theory
61:
59:Thermodynamics
53:
52:
44:
43:
26:
9:
6:
4:
3:
2:
4111:
4100:
4097:
4095:
4092:
4090:
4087:
4086:
4084:
4067:
4064:
4062:
4059:
4057:
4054:
4053:
4052:
4049:
4047:
4044:
4040:
4039:superfluidity
4037:
4035:
4032:
4031:
4030:
4027:
4026:
4024:
4020:
4014:
4011:
4009:
4006:
4004:
4001:
3999:
3996:
3994:
3991:
3990:
3988:
3986:
3982:
3974:
3971:
3969:
3966:
3965:
3964:
3961:
3959:
3956:
3955:
3953:
3951:
3947:
3941:
3937:
3934:
3932:
3929:
3927:
3924:
3922:
3919:
3917:
3914:
3912:
3909:
3908:
3906:
3902:
3894:
3891:
3889:
3886:
3885:
3884:
3880:
3876:
3873:
3871:
3868:
3866:
3863:
3861:
3858:
3857:
3856:
3853:
3852:
3850:
3848:
3844:
3838:
3835:
3831:
3828:
3826:
3823:
3821:
3818:
3816:
3813:
3812:
3810:
3807:
3805:
3802:
3800:
3797:
3795:
3792:
3791:
3789:
3787:
3783:
3768:
3765:
3763:
3760:
3759:
3757:
3753:
3749:
3742:
3737:
3735:
3730:
3728:
3723:
3722:
3719:
3713:
3710:
3709:
3689:
3682:
3676:
3662:
3658:
3653:
3648:
3645:(5): 051118.
3644:
3640:
3636:
3629:
3621:
3617:
3613:
3609:
3605:
3601:
3593:
3585:
3578:
3570:
3568:0-08-023038-5
3564:
3560:
3556:
3550:
3542:
3540:0-7167-1088-9
3536:
3532:
3528:
3522:
3514:
3510:
3509:
3504:
3498:
3496:
3494:
3492:
3490:
3488:
3486:
3484:
3482:
3480:
3471:
3465:
3461:
3454:
3450:
3436:
3429:
3420:
3413:
3408:
3405:) which is a
3404:
3401:(e. g., unit
3400:
3396:
3390:
3383:
3378:
3371:
3364:
3357:
3353:
3347:
3343:
3333:
3330:
3328:
3325:
3323:
3320:
3318:
3315:
3313:
3310:
3308:
3305:
3304:
3298:
3281:
3275:
3272:
3266:
3263:
3260:
3254:
3248:
3241:
3240:
3239:
3237:
3233:
3229:
3228:quantum logic
3205:
3201:
3197:
3194:
3188:
3185:
3180:
3175:
3172:
3169:
3165:
3159:
3156:
3143:
3135:
3129:
3123:
3116:
3115:
3114:
3111:
3108:
3104:
3100:
3093:
3089:
3082:
3078:
3074:
3070:
3066:
3062:
3058:
3053:
3051:
3046:
3042:
3035:
3028:
3023:
3018:
3012:
3009:
3007:
3002:
2998:
2997:
2996:
2994:
2983:
2981:
2977:
2973:
2969:
2965:
2961:
2957:
2953:
2949:
2945:
2941:
2937:
2932:
2930:
2926:
2922:
2918:
2894:
2884:
2880:
2876:
2873:
2869:
2863:
2859:
2849:
2845:
2841:
2838:
2834:
2828:
2824:
2818:
2814:
2807:
2798:
2788:
2787:
2786:
2784:
2780:
2769:
2766:
2758:
2755:November 2023
2748:
2744:
2740:
2734:
2733:
2729:
2724:This section
2722:
2718:
2713:
2712:
2704:
2702:
2698:
2690:
2674:
2671:
2664:
2647:
2643:
2620:
2616:
2607:
2603:
2600:
2597:
2578:
2575:
2571:
2549:
2542:
2539:
2517:
2507:
2506:
2505:
2488:
2481:
2478:
2467:
2463:
2459:
2456:
2453:
2448:
2444:
2440:
2435:
2431:
2427:
2422:
2418:
2414:
2411:
2408:
2403:
2399:
2395:
2390:
2386:
2379:
2376:
2373:
2369:
2364:
2358:
2355:
2344:
2340:
2336:
2333:
2330:
2325:
2321:
2317:
2312:
2308:
2304:
2299:
2295:
2291:
2288:
2285:
2280:
2276:
2272:
2267:
2263:
2256:
2253:
2250:
2246:
2242:
2238:
2232:
2223:
2213:
2212:
2211:
2209:
2205:
2201:
2191:
2189:
2185:
2181:
2177:
2173:
2168:
2166:
2162:
2158:
2154:
2150:
2140:
2138:
2134:
2130:
2126:
2122:
2118:
2114:
2110:
2105:
2103:
2099:
2095:
2091:
2090:Gibbs measure
2087:
2081:
2077:
2067:
2050:
2047:
2042:
2038:
2034:
2031:
2026:
2022:
2018:
2013:
2009:
2005:
2002:
1995:
1994:
1993:
1989:
1986:
1979:
1972:
1968:
1962:
1955:
1948:
1941:
1937:
1933:
1930:
1926:
1923:
1922:
1921:
1912:
1909:
1903:
1897:
1887:
1883:
1879:
1869:
1865:
1864:
1863:
1846:
1843:
1837:
1832:
1828:
1823:
1818:
1815:
1808:
1807:
1806:
1803:
1797:
1791:
1787:
1768:
1765:
1760:
1756:
1752:
1749:
1744:
1740:
1736:
1732:
1729:
1726:
1723:
1713:
1710:
1705:
1701:
1696:
1692:
1682:
1679:
1674:
1670:
1665:
1657:
1656:
1655:
1638:
1633:
1629:
1625:
1622:
1617:
1613:
1609:
1605:
1602:
1599:
1596:
1593:
1583:
1580:
1575:
1571:
1566:
1562:
1552:
1549:
1544:
1540:
1535:
1531:
1525:
1515:
1514:
1513:
1510:
1504:
1498:
1495:
1487:
1483:
1476:
1471:
1467:
1460:
1455:
1451:
1444:
1440:
1436:
1431:
1424:
1420:
1411:
1402:
1395:
1390:
1386:
1379:
1374:
1370:
1363:
1359:
1355:
1349:
1345:
1338:
1333:
1328:
1321:
1317:
1310:
1305:
1301:
1296:
1291:
1289:
1285:
1281:
1273:
1269:
1265:
1260:
1251:
1246:, indexed by
1242:
1238:
1217:
1207:
1203:
1191:
1187:
1176:
1172:
1166:
1162:
1158:
1149:
1139:
1138:
1137:
1135:
1131:
1109:
1080:
1069:
1065:
1049:
1046:
1043:
1040:
1036:
1026:
1020:
1011:
1009:
1004:
990:
987:
978:
972:
969:
943:
931:
920:
902:
892:
880:
858:
832:
826:
817:
808:
805:
802:
796:
786:
785:
784:
782:
766:
740:
707:
697:, denoted by
696:
690:
680:
678:
669:
663:
659:
655:
649:
643:
639:
636:
632:
629:Test whether
628:
627:
626:
624:
620:
616:
615:operationally
606:
604:
600:
596:
592:
581:
579:
569:
567:
563:
562:stoichiometry
553:
549:
545:
544:
540:
536:
532:
528:
524:
523:
519:
516:
512:
511:
507:
506:
505:
501:
499:
494:
488:
484:
480:
476:
472:
467:
455:
451:
447:
444:
440:
436:
432:
431:
425:
423:
419:
413:
411:
406:
402:
398:
393:
384:
382:
378:
373:
371:
367:
363:
359:
355:
351:
339:
334:
332:
327:
325:
320:
319:
317:
316:
309:
306:
304:
301:
299:
296:
294:
291:
289:
286:
284:
281:
279:
276:
274:
271:
269:
266:
264:
261:
260:
254:
253:
246:
243:
241:
238:
236:
233:
231:
228:
226:
223:
222:
219:
214:
213:
206:
203:
201:
198:
196:
193:
191:
188:
187:
181:
180:
173:
170:
167:
165:
162:
159:
157:
154:
151:
149:
146:
143:
141:
138:
135:
134:
131:
126:
125:
118:
115:
113:
110:
108:
105:
103:
100:
98:
97:BoseâEinstein
95:
93:
90:
88:
85:
83:
80:
79:
76:
71:
70:
65:
62:
60:
57:
56:
55:
54:
50:
46:
45:
42:
39:
38:
33:
19:
4022:Applications
3973:size scaling
3691:. Retrieved
3687:
3675:
3664:. Retrieved
3642:
3638:
3628:
3603:
3599:
3592:
3583:
3577:
3558:
3555:Landau, L.D.
3549:
3530:
3521:
3511:. New York:
3506:
3459:
3453:
3428:
3418:
3411:
3407:double cover
3389:
3376:
3369:
3363:
3358:, Chapter I.
3355:
3346:
3296:
3235:
3231:
3225:
3112:
3106:
3102:
3098:
3091:
3087:
3080:
3076:
3072:
3068:
3064:
3060:
3056:
3054:
3044:
3040:
3033:
3026:
3021:
3019:
3016:
3005:
3000:
2992:
2989:
2979:
2975:
2971:
2963:
2959:
2955:
2947:
2943:
2939:
2933:
2914:
2776:
2761:
2752:
2737:Please help
2725:
2700:
2694:
2601:
2537:
2503:
2203:
2197:
2169:
2152:
2146:
2106:
2083:
2065:
1990:
1984:
1977:
1973:
1966:
1960:
1953:
1949:
1945:
1918:
1907:
1901:
1895:
1892:
1885:
1877:
1867:
1861:
1805:by a factor
1801:
1795:
1790:distribution
1789:
1785:
1783:
1653:
1508:
1502:
1499:
1493:
1485:
1481:
1474:
1469:
1465:
1458:
1453:
1449:
1442:
1438:
1429:
1422:
1418:
1409:
1400:
1396:
1388:
1384:
1377:
1372:
1368:
1361:
1357:
1347:
1343:
1336:
1326:
1319:
1315:
1308:
1299:
1292:
1277:
1240:
1236:
1232:
1012:
1005:
847:
692:
674:
667:
661:
657:
653:
647:
641:
634:
630:
622:
618:
612:
587:
575:
558:
548:ÎŒVT ensemble
547:
541:
538:temperature.
527:NVT ensemble
526:
520:
515:NVE ensemble
514:
508:
503:
497:
496:
491:
421:
417:
414:
394:
390:
376:
374:
361:
357:
347:
168:
160:
152:
144:
136:
4013:von Neumann
3883:force field
3875:percolation
3403:quaternions
3317:Phase space
3059:subsequent
2942:), volume (
2606:Hamiltonian
2594:, known as
2208:phase space
2188:open system
2117:Ising model
2098:linguistics
1873:energyĂtime
1330:associated
1280:phase space
1268:phase space
1266:systems in
919:covariances
603:phase space
595:von Neumann
572:Equivalence
531:temperature
452:, the term
428:Terminology
405:macroscopic
293:von Neumann
102:FermiâDirac
4083:Categories
3870:Heisenberg
3693:3 November
3666:2024-03-03
3445:References
2161:microstate
1233:where the
461:Main types
418:stationary
257:Scientists
218:Potentials
3993:Boltzmann
3916:H-theorem
3794:Ensembles
3652:0810.3407
3620:0892-7022
3267:
3249:σ
3238:so that:
3189:
3166:∑
3150:∞
3147:→
3124:σ
3086:), Meas (
2877:β
2874:−
2860:∑
2842:β
2839:−
2815:∑
2802:¯
2726:does not
2675:τ
2550:β
2521:¯
2482:τ
2457:…
2412:…
2377:β
2374:−
2365:∫
2359:τ
2334:…
2289:…
2254:β
2251:−
2239:∫
2227:¯
2035:…
1816:ρ
1750:…
1733:ρ
1730:∫
1727:…
1724:∫
1719:∞
1697:∑
1693:…
1688:∞
1666:∑
1623:…
1603:ρ
1600:∫
1597:…
1594:∫
1589:∞
1567:∑
1563:…
1558:∞
1536:∑
1529:⟩
1523:⟨
1264:classical
1204:ψ
1200:⟨
1197:⟩
1188:ψ
1163:∑
1153:^
1150:ρ
1113:^
1084:^
1030:^
1027:ρ
982:^
979:ρ
973:
947:^
935:^
896:^
879:variances
862:^
827:ρ
821:^
809:
800:⟩
794:⟨
767:ρ
744:^
711:^
708:ρ
660:and from
372:in 1902.
288:Ehrenfest
268:Boltzmann
148:Canonical
4003:Tsallis
3505:(1902).
3399:3-sphere
3301:See also
3050:infinite
2172:ensemble
2165:ensemble
2119:, model
2102:robotics
1938:and the
358:ensemble
283:Einstein
230:Enthalpy
195:Einstein
3998:Shannon
3985:Entropy
3435:measure
2747:removed
2732:sources
2687:is the
2604:is the
1786:density
1334:called
1306:called
646:0 <
564:of the
439:walkers
435:sampled
350:physics
263:Maxwell
3847:Models
3755:Theory
3618:
3565:
3537:
3466:
3232:yesâno
3039:, ...,
2993:itself
2504:where
2182:. The
2151:, the
1893:Since
1862:where
1480:, ...
1464:, ...
1448:, ...
1383:, ...
1367:, ...
1342:, ...
1324:, and
1314:, ...
650:< 1
500:(1903)
366:system
360:(also
298:Tolman
184:Models
4056:chaos
4008:RĂ©nyi
3865:Potts
3860:Ising
3684:(PDF)
3647:arXiv
3421:log 2
3414:log 2
3372:= 1 Ă
3338:Notes
1293:In a
781:trace
441:in a
356:, an
308:Fermi
303:Debye
278:Gibbs
205:Potts
200:Ising
190:Debye
3938:and
3695:2023
3616:ISSN
3563:ISBN
3535:ISBN
3464:ISBN
3186:Meas
2934:The
2927:and
2730:any
2728:cite
2176:mean
2157:mean
2078:and
1066:and
666:1 â
546:(or
525:(or
513:(or
273:Bose
3657:doi
3608:doi
3140:lim
3067:or
3065:yes
3057:one
3006:gas
2777:In
2741:by
2562:is
2147:In
2088:or
1969:= 1
917:),
877:),
640:If
348:In
169:NPT
161:NPH
153:”VT
145:NVT
137:NVE
4085::
3811::
3686:.
3655:.
3643:79
3641:.
3637:.
3614:.
3604:34
3602:.
3478:^
3419:kT
3264:Tr
3101:,
3090:,
3079:,
3069:no
3032:,
3010:?)
2931:.
2923:,
2703:.
2190:.
2167:.
2139:.
2100:,
1769:1.
1512::
1473:,
1457:,
1394:.
1376:,
1290:.
1010:.
970:Tr
806:Tr
783::
633:,
621:,
485:,
481:,
477:,
473:,
424:.
412:.
375:A
3830:G
3825:F
3820:H
3815:U
3740:e
3733:t
3726:v
3697:.
3669:.
3659::
3649::
3622:.
3610::
3571:.
3543:.
3515:.
3472:.
3412:k
3377:h
3370:h
3282:.
3279:)
3276:S
3273:E
3270:(
3261:=
3258:)
3255:E
3252:(
3236:S
3211:)
3206:k
3202:X
3198:,
3195:E
3192:(
3181:N
3176:1
3173:=
3170:k
3160:N
3157:1
3144:N
3136:=
3133:)
3130:E
3127:(
3107:k
3103:X
3099:E
3095:2
3092:X
3088:E
3084:1
3081:X
3077:E
3073:E
3045:k
3041:X
3037:2
3034:X
3030:1
3027:X
2980:V
2976:N
2972:E
2964:N
2960:V
2956:E
2948:N
2944:V
2940:E
2895:.
2885:i
2881:E
2870:e
2864:i
2850:i
2846:E
2835:e
2829:i
2825:A
2819:i
2808:=
2799:A
2768:)
2762:(
2757:)
2753:(
2749:.
2735:.
2701:Z
2672:d
2662:,
2648:i
2644:p
2621:i
2617:q
2602:H
2598:,
2579:T
2576:k
2572:1
2540:,
2538:A
2518:A
2489:,
2479:d
2473:)
2468:N
2464:p
2460:,
2454:,
2449:2
2445:p
2441:,
2436:1
2432:p
2428:,
2423:M
2419:q
2415:,
2409:,
2404:2
2400:q
2396:,
2391:1
2387:q
2383:(
2380:H
2370:e
2356:d
2350:)
2345:N
2341:p
2337:,
2331:,
2326:2
2322:p
2318:,
2313:1
2309:p
2305:,
2300:M
2296:q
2292:,
2286:,
2281:2
2277:q
2273:,
2268:1
2264:q
2260:(
2257:H
2247:e
2243:A
2233:=
2224:A
2051:.
2048:!
2043:s
2039:N
2032:!
2027:2
2023:N
2019:!
2014:1
2010:N
2006:=
2003:C
1985:C
1978:C
1967:C
1961:C
1954:x
1942:.
1931:.
1908:h
1902:h
1896:h
1886:C
1881:.
1878:Ï
1868:h
1847:,
1844:P
1838:C
1833:n
1829:h
1824:1
1819:=
1802:P
1796:Ï
1766:=
1761:n
1757:q
1753:d
1745:1
1741:p
1737:d
1714:0
1711:=
1706:s
1702:N
1683:0
1680:=
1675:1
1671:N
1639:.
1634:n
1630:q
1626:d
1618:1
1614:p
1610:d
1606:X
1584:0
1581:=
1576:s
1572:N
1553:0
1550:=
1545:1
1541:N
1532:=
1526:X
1509:Ï
1503:X
1494:n
1489:)
1486:n
1482:q
1478:1
1475:q
1470:n
1466:p
1462:1
1459:p
1454:s
1450:N
1446:1
1443:N
1441:(
1439:Ï
1430:s
1423:s
1419:N
1413:2
1410:N
1404:1
1401:N
1392:)
1389:n
1385:q
1381:1
1378:q
1373:n
1369:p
1365:1
1362:p
1360:(
1358:Ï
1348:n
1344:p
1340:1
1337:p
1327:n
1320:n
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1241:i
1237:Ï
1235:|
1218:,
1214:|
1208:i
1192:i
1183:|
1177:i
1173:P
1167:i
1159:=
1110:N
1081:H
1050:0
1047:=
1044:t
1041:d
1037:/
1021:d
991:1
988:=
944:Y
932:X
903:2
893:X
859:X
833:.
830:)
818:X
812:(
803:=
797:X
741:X
728:X
671:.
668:p
662:B
658:p
654:A
648:p
642:p
635:B
631:A
623:B
619:A
337:e
330:t
323:v
34:.
20:)
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