1100:
690:
440:
for the full nonlinear response. If the system in question is highly non-linear, higher order terms in the expansion, denoted by the dots, become important and the signal transducer cannot adequately be described just by its linear response function.
959:
428:
1490:
946:
570:
790:
1351:
743:
1244:
1146:
1698:
1656:
1207:
483:
543:
1611:
316:
1274:
1568:
1304:
1519:
276:
1169:
563:
838:
831:
242:
213:
184:
155:
122:
321:
1816:
in Eva
Pavarini, Erik Koch, Dieter Vollhardt, and Alexander Lichtenstein (eds.): DMFT at 25: Infinite Dimensions, Verlag des Forschungszentrum JĂĽlich, 2014
1095:{\displaystyle {\tilde {\chi }}(\omega )={\frac {{\tilde {x}}(\omega )}{{\tilde {h}}(\omega )}}={\frac {1}{\omega _{0}^{2}-\omega ^{2}+i\gamma \omega }}.}
1813:
1391:
1821:
751:
485:
of the linear response function is very useful as it describes the output of the system if the input is a sine wave
1312:
1571:
700:
17:
1216:
1110:
685:{\displaystyle x(t)=\left|{\tilde {\chi }}(\omega )\right|h_{0}\sin(\omega t+\arg {\tilde {\chi }}(\omega ))\,,}
1665:
1623:
1174:
450:
84:
488:
1836:
1659:
1701:
801:
1253:
1718:
1535:
1246:. The linear response function for a harmonic oscillator is mathematically identical to that of an
1148:
and the phase shift by the arctan of the imaginary part of the function divided by the real one.
1279:
1576:
281:
60:
1495:
1154:
548:
1385:
247:
80:
68:
36:
807:
218:
189:
160:
131:
98:
8:
1753:
1743:
1738:
1728:
1723:
1617:
76:
1362:
48:
1817:
1733:
951:
445:
72:
32:
1748:
64:
437:
1307:
1105:
1532:
is the perturbation of the thermal expectation of another measurable quantity
1830:
941:{\displaystyle {\ddot {x}}(t)+\gamma {\dot {x}}(t)+\omega _{0}^{2}x(t)=h(t).}
433:
59:
there exist alternative names for specific linear response functions such as
1570:. The Kubo formula then defines the quantum-statistical calculation of the
1370:
1713:
1247:
746:
423:{\displaystyle x(t)=\int _{-\infty }^{t}dt'\,\chi (t-t')h(t')+\cdots \,.}
56:
1758:
1366:
1210:
695:
1485:{\displaystyle {\hat {H}}_{0}\to {\hat {H}}_{0}-h(t'){\hat {B}}(t')}
52:
44:
1521:
corresponds to a measurable quantity as input, while the output
40:
1613:
by a general formula involving only the mentioned operators.
1209:
of the linear response function yields a pronounced maximum ("
125:
1384:
is a perturbation of the basic operator of the system, the
1361:
The exposition of linear response theory, in the context of
47:
input into a response. Because of its many applications in
1658:
has poles only in the lower half-plane. This leads to the
1778:
Statistical
Mechanical Theory of Irreversible Processes I
278:, with the weights given by the linear response function
1700:
by integration. The simplest example is once more the
1662:, which relates the real and the imaginary parts of
1668:
1626:
1579:
1538:
1498:
1394:
1315:
1282:
1256:
1219:
1177:
1157:
1113:
962:
841:
810:
754:
703:
573:
551:
491:
453:
324:
284:
250:
221:
192:
163:
134:
101:
1373:, which considers the general case that the "force"
1104:
The amplitude gain is given by the magnitude of the
1692:
1650:
1605:
1562:
1513:
1484:
1345:
1298:
1268:
1238:
1201:
1163:
1140:
1094:
940:
825:
784:
737:
684:
557:
537:
477:
422:
310:
270:
236:
207:
178:
149:
116:
1780:, Journal of the Physical Society of Japan, vol.
1828:
1151:From this representation, we see that for small
432:The explicit term on the right-hand side is the
804:with input given by an external driving force
785:{\displaystyle \arg {\tilde {\chi }}(\omega )}
1346:{\displaystyle Q:=\omega _{0}/\Delta \omega }
186:will depend not only on the present value of
31:describes the input-output relationship of a
954:of the linear response function is given by
244:is a weighted sum of the previous values of
738:{\displaystyle |{\tilde {\chi }}(\omega )|}
157:(e.g. a position). Generally, the value of
1239:{\displaystyle \omega \approx \omega _{0}}
1141:{\displaystyle {\tilde {\chi }}(\omega ),}
90:
1693:{\displaystyle {\tilde {\chi }}(\omega )}
1651:{\displaystyle {\tilde {\chi }}(\omega )}
1202:{\displaystyle {\tilde {\chi }}(\omega )}
678:
478:{\displaystyle {\tilde {\chi }}(\omega )}
416:
369:
215:, but also on past values. Approximately
538:{\displaystyle h(t)=h_{0}\sin(\omega t)}
14:
1829:
1616:As a consequence of the principle of
128:), and the response of the system by
24:
1369:. This defines particularly the
1337:
1257:
348:
25:
1848:
1807:
95:Denote the input of a system by
1356:
1276:typically is much smaller than
1269:{\displaystyle \Delta \omega ,}
1787:
1770:
1687:
1681:
1675:
1645:
1639:
1633:
1600:
1583:
1557:
1551:
1545:
1505:
1479:
1468:
1462:
1453:
1442:
1424:
1414:
1402:
1196:
1190:
1184:
1132:
1126:
1120:
1031:
1025:
1019:
1008:
1002:
996:
981:
975:
969:
932:
926:
917:
911:
887:
881:
860:
854:
820:
814:
779:
773:
767:
731:
727:
721:
715:
705:
675:
672:
666:
660:
636:
612:
606:
600:
583:
577:
532:
523:
501:
495:
472:
466:
460:
407:
396:
390:
373:
334:
328:
305:
288:
265:
254:
231:
225:
202:
196:
173:
167:
144:
138:
111:
105:
85:ordinary differential equation
13:
1:
1764:
1563:{\displaystyle {\hat {A}}(t)}
1365:, can be found in a paper by
1620:the complex-valued function
1299:{\displaystyle \omega _{0},}
1250:. The width of the maximum,
7:
1707:
1606:{\displaystyle \chi (t-t')}
311:{\displaystyle \chi (t-t')}
10:
1853:
1799:Theory of condensed matter
1797:, in: E. AntonÄŤik et al.,
1702:damped harmonic oscillator
1514:{\displaystyle {\hat {B}}}
802:damped harmonic oscillator
795:
35:, such as a radio turning
1814:Linear Response Functions
1353:can be extremely large.
1660:Kramers–Kronig relations
29:linear response function
1164:{\displaystyle \gamma }
558:{\displaystyle \omega }
91:Mathematical definition
1795:Linear Response Theory
1694:
1652:
1607:
1564:
1515:
1486:
1347:
1300:
1270:
1240:
1203:
1171:the Fourier transform
1165:
1142:
1096:
942:
827:
786:
739:
686:
559:
539:
479:
424:
312:
272:
238:
209:
180:
151:
118:
1784:, pp. 570–586 (1957).
1695:
1653:
1608:
1565:
1516:
1487:
1348:
1301:
1271:
1241:
1204:
1166:
1143:
1097:
943:
828:
787:
740:
687:
560:
540:
480:
425:
313:
273:
271:{\displaystyle h(t')}
239:
210:
181:
152:
119:
37:electromagnetic waves
1837:Equations of physics
1719:Green–Kubo relations
1666:
1624:
1577:
1536:
1496:
1392:
1313:
1280:
1254:
1217:
1213:") at the frequency
1175:
1155:
1111:
960:
839:
826:{\displaystyle h(t)}
808:
752:
701:
571:
549:
489:
451:
322:
282:
248:
237:{\displaystyle x(t)}
219:
208:{\displaystyle h(t)}
190:
179:{\displaystyle x(t)}
161:
150:{\displaystyle x(t)}
132:
117:{\displaystyle h(t)}
99:
87:is closely related.
81:fundamental solution
1801:, IAEA Vienna, 1968
1754:Resolvent formalism
1739:Semilinear response
1729:Dispersion (optics)
1724:Fluctuation theorem
1060:
950:The complex-valued
907:
565:. The output reads
444:The complex-valued
357:
75:. The concept of a
1690:
1648:
1603:
1560:
1511:
1482:
1363:quantum statistics
1343:
1296:
1266:
1236:
1199:
1161:
1138:
1092:
1046:
938:
893:
823:
782:
735:
682:
555:
535:
475:
438:Volterra expansion
420:
340:
308:
268:
234:
205:
176:
147:
114:
49:information theory
1822:978-3-89336-953-9
1734:Lindblad equation
1678:
1636:
1548:
1508:
1465:
1427:
1405:
1187:
1123:
1087:
1035:
1022:
999:
972:
952:Fourier transform
878:
851:
770:
718:
663:
603:
463:
446:Fourier transform
73:transfer function
33:signal transducer
16:(Redirected from
1844:
1802:
1791:
1785:
1774:
1749:Impulse response
1744:Green's function
1699:
1697:
1696:
1691:
1680:
1679:
1671:
1657:
1655:
1654:
1649:
1638:
1637:
1629:
1612:
1610:
1609:
1604:
1599:
1569:
1567:
1566:
1561:
1550:
1549:
1541:
1531:
1520:
1518:
1517:
1512:
1510:
1509:
1501:
1491:
1489:
1488:
1483:
1478:
1467:
1466:
1458:
1452:
1435:
1434:
1429:
1428:
1420:
1413:
1412:
1407:
1406:
1398:
1383:
1352:
1350:
1349:
1344:
1336:
1331:
1330:
1305:
1303:
1302:
1297:
1292:
1291:
1275:
1273:
1272:
1267:
1245:
1243:
1242:
1237:
1235:
1234:
1208:
1206:
1205:
1200:
1189:
1188:
1180:
1170:
1168:
1167:
1162:
1147:
1145:
1144:
1139:
1125:
1124:
1116:
1101:
1099:
1098:
1093:
1088:
1086:
1073:
1072:
1059:
1054:
1041:
1036:
1034:
1024:
1023:
1015:
1011:
1001:
1000:
992:
988:
974:
973:
965:
947:
945:
944:
939:
906:
901:
880:
879:
871:
853:
852:
844:
832:
830:
829:
824:
791:
789:
788:
783:
772:
771:
763:
744:
742:
741:
736:
734:
720:
719:
711:
708:
691:
689:
688:
683:
665:
664:
656:
629:
628:
619:
615:
605:
604:
596:
564:
562:
561:
556:
544:
542:
541:
536:
516:
515:
484:
482:
481:
476:
465:
464:
456:
429:
427:
426:
421:
406:
389:
368:
356:
351:
317:
315:
314:
309:
304:
277:
275:
274:
269:
264:
243:
241:
240:
235:
214:
212:
211:
206:
185:
183:
182:
177:
156:
154:
153:
148:
123:
121:
120:
115:
77:Green's function
65:impulse response
39:into music or a
21:
1852:
1851:
1847:
1846:
1845:
1843:
1842:
1841:
1827:
1826:
1810:
1805:
1792:
1788:
1775:
1771:
1767:
1710:
1670:
1669:
1667:
1664:
1663:
1628:
1627:
1625:
1622:
1621:
1592:
1578:
1575:
1574:
1540:
1539:
1537:
1534:
1533:
1522:
1500:
1499:
1497:
1494:
1493:
1471:
1457:
1456:
1445:
1430:
1419:
1418:
1417:
1408:
1397:
1396:
1395:
1393:
1390:
1389:
1374:
1359:
1332:
1326:
1322:
1314:
1311:
1310:
1287:
1283:
1281:
1278:
1277:
1255:
1252:
1251:
1230:
1226:
1218:
1215:
1214:
1179:
1178:
1176:
1173:
1172:
1156:
1153:
1152:
1115:
1114:
1112:
1109:
1108:
1068:
1064:
1055:
1050:
1045:
1040:
1014:
1013:
1012:
991:
990:
989:
987:
964:
963:
961:
958:
957:
902:
897:
870:
869:
843:
842:
840:
837:
836:
809:
806:
805:
798:
762:
761:
753:
750:
749:
730:
710:
709:
704:
702:
699:
698:
655:
654:
624:
620:
595:
594:
593:
589:
572:
569:
568:
550:
547:
546:
545:with frequency
511:
507:
490:
487:
486:
455:
454:
452:
449:
448:
399:
382:
361:
352:
344:
323:
320:
319:
297:
283:
280:
279:
257:
249:
246:
245:
220:
217:
216:
191:
188:
187:
162:
159:
158:
133:
130:
129:
100:
97:
96:
93:
23:
22:
18:Linear response
15:
12:
11:
5:
1850:
1840:
1839:
1825:
1824:
1809:
1808:External links
1806:
1804:
1803:
1786:
1768:
1766:
1763:
1762:
1761:
1756:
1751:
1746:
1741:
1736:
1731:
1726:
1721:
1716:
1709:
1706:
1689:
1686:
1683:
1677:
1674:
1647:
1644:
1641:
1635:
1632:
1602:
1598:
1595:
1591:
1588:
1585:
1582:
1572:susceptibility
1559:
1556:
1553:
1547:
1544:
1507:
1504:
1481:
1477:
1474:
1470:
1464:
1461:
1455:
1451:
1448:
1444:
1441:
1438:
1433:
1426:
1423:
1416:
1411:
1404:
1401:
1358:
1355:
1342:
1339:
1335:
1329:
1325:
1321:
1318:
1308:Quality factor
1295:
1290:
1286:
1265:
1262:
1259:
1233:
1229:
1225:
1222:
1198:
1195:
1192:
1186:
1183:
1160:
1137:
1134:
1131:
1128:
1122:
1119:
1106:complex number
1091:
1085:
1082:
1079:
1076:
1071:
1067:
1063:
1058:
1053:
1049:
1044:
1039:
1033:
1030:
1027:
1021:
1018:
1010:
1007:
1004:
998:
995:
986:
983:
980:
977:
971:
968:
937:
934:
931:
928:
925:
922:
919:
916:
913:
910:
905:
900:
896:
892:
889:
886:
883:
877:
874:
868:
865:
862:
859:
856:
850:
847:
822:
819:
816:
813:
797:
794:
781:
778:
775:
769:
766:
760:
757:
733:
729:
726:
723:
717:
714:
707:
696:amplitude gain
681:
677:
674:
671:
668:
662:
659:
653:
650:
647:
644:
641:
638:
635:
632:
627:
623:
618:
614:
611:
608:
602:
599:
592:
588:
585:
582:
579:
576:
554:
534:
531:
528:
525:
522:
519:
514:
510:
506:
503:
500:
497:
494:
474:
471:
468:
462:
459:
419:
415:
412:
409:
405:
402:
398:
395:
392:
388:
385:
381:
378:
375:
372:
367:
364:
360:
355:
350:
347:
343:
339:
336:
333:
330:
327:
307:
303:
300:
296:
293:
290:
287:
267:
263:
260:
256:
253:
233:
230:
227:
224:
204:
201:
198:
195:
175:
172:
169:
166:
146:
143:
140:
137:
113:
110:
107:
104:
92:
89:
61:susceptibility
9:
6:
4:
3:
2:
1849:
1838:
1835:
1834:
1832:
1823:
1819:
1815:
1812:
1811:
1800:
1796:
1790:
1783:
1779:
1773:
1769:
1760:
1757:
1755:
1752:
1750:
1747:
1745:
1742:
1740:
1737:
1735:
1732:
1730:
1727:
1725:
1722:
1720:
1717:
1715:
1712:
1711:
1705:
1703:
1684:
1672:
1661:
1642:
1630:
1619:
1614:
1596:
1593:
1589:
1586:
1580:
1573:
1554:
1542:
1529:
1525:
1502:
1475:
1472:
1459:
1449:
1446:
1439:
1436:
1431:
1421:
1409:
1399:
1387:
1381:
1377:
1372:
1368:
1364:
1354:
1340:
1333:
1327:
1323:
1319:
1316:
1309:
1293:
1288:
1284:
1263:
1260:
1249:
1231:
1227:
1223:
1220:
1212:
1193:
1181:
1158:
1149:
1135:
1129:
1117:
1107:
1102:
1089:
1083:
1080:
1077:
1074:
1069:
1065:
1061:
1056:
1051:
1047:
1042:
1037:
1028:
1016:
1005:
993:
984:
978:
966:
955:
953:
948:
935:
929:
923:
920:
914:
908:
903:
898:
894:
890:
884:
875:
872:
866:
863:
857:
848:
845:
834:
817:
811:
803:
793:
776:
764:
758:
755:
748:
724:
712:
697:
692:
679:
669:
657:
651:
648:
645:
642:
639:
633:
630:
625:
621:
616:
609:
597:
590:
586:
580:
574:
566:
552:
529:
526:
520:
517:
512:
508:
504:
498:
492:
469:
457:
447:
442:
439:
435:
434:leading order
430:
417:
413:
410:
403:
400:
393:
386:
383:
379:
376:
370:
365:
362:
358:
353:
345:
341:
337:
331:
325:
301:
298:
294:
291:
285:
261:
258:
251:
228:
222:
199:
193:
170:
164:
141:
135:
127:
108:
102:
88:
86:
82:
78:
74:
70:
66:
62:
58:
54:
50:
46:
42:
38:
34:
30:
19:
1798:
1794:
1793:De Clozeaux,
1789:
1781:
1777:
1772:
1615:
1527:
1523:
1379:
1375:
1371:Kubo formula
1360:
1357:Kubo formula
1306:so that the
1150:
1103:
956:
949:
835:
799:
693:
567:
443:
431:
94:
28:
26:
1714:Convolution
1386:Hamiltonian
1248:RLC circuit
800:Consider a
747:phase shift
71:; see also
57:engineering
1776:Kubo, R.,
1765:References
1759:Propagator
1367:Ryogo Kubo
436:term of a
1685:ω
1676:~
1673:χ
1643:ω
1634:~
1631:χ
1618:causality
1590:−
1581:χ
1546:^
1506:^
1463:^
1437:−
1425:^
1415:→
1403:^
1341:ω
1338:Δ
1324:ω
1285:ω
1261:ω
1258:Δ
1228:ω
1224:≈
1221:ω
1211:Resonance
1194:ω
1185:~
1182:χ
1159:γ
1130:ω
1121:~
1118:χ
1084:ω
1081:γ
1066:ω
1062:−
1048:ω
1029:ω
1020:~
1006:ω
997:~
979:ω
970:~
967:χ
895:ω
876:˙
867:γ
849:¨
777:ω
768:~
765:χ
759:
725:ω
716:~
713:χ
670:ω
661:~
658:χ
652:
640:ω
634:
610:ω
601:~
598:χ
553:ω
527:ω
521:
470:ω
461:~
458:χ
414:⋯
380:−
371:χ
349:∞
346:−
342:∫
295:−
286:χ
69:impedance
1831:Category
1708:See also
1597:′
1476:′
1450:′
404:′
387:′
366:′
302:′
262:′
124:(e.g. a
45:synaptic
43:turning
796:Example
53:physics
1820:
1492:where
83:of an
41:neuron
694:with
126:force
1818:ISBN
745:and
55:and
756:arg
649:arg
631:sin
518:sin
79:or
67:or
1833::
1782:12
1704:.
1388:,
1320::=
833:,
792:.
318::
63:,
51:,
27:A
1688:)
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1090:.
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1075:+
1070:2
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109:t
106:(
103:h
20:)
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