Ergodic hypothesis

The ergodic hypothesis (often also referred to as the ergodic theorem ) states that thermodynamic systems usually behave randomly (“ molecular chaos ”), so that all energetically possible phase space regions are also reached. The time span during which a trajectory in the phase space of the microstates is in a certain region is proportional to the volume of this region. In other words, the hypothesis says that thermodynamic systems have the property of ergodicity .

The ergodic hypothesis is fundamental to statistical mechanics . Among other things, it combines the results of molecular dynamics simulations and Monte Carlo simulations .

Definition and restriction

It is precisely assumed that the time average is equal to the ensemble average for almost all measured variables : ${\ displaystyle A}$ ${\ displaystyle {\ overline {A (t)}}}$ ${\ displaystyle \ langle A \ rangle}$

{\ displaystyle {\ overline {A (t)}} = \ lim _ {t \ to \ infty} {\ frac {1} {t}} \ int _ {0} ^ {t} A (t ') \ mathrm {d} t '{\ overset {!} {=}} \ langle A \ rangle = \ sum _ {i} p_ {i} A_ {i},}

where is the probability of the state given by the probability distribution of the ensemble . ${\ displaystyle p_ {i}}$ ${\ displaystyle i}$

The prerequisite for validity is that the stochastic process under consideration is stationary and has a finite correlation time; then the ergodic hypothesis applies in the Limes of infinite time.

Furthermore, a dynamic system is ergodic (more precisely: quasi-ergodic ) insofar as the trajectory (i.e. the path of the system) comes arbitrarily close to any point in phase space in finite time . On the other hand, Ludwig Boltzmann formulated in his original work in 1887 that the train hits every point .

Although the ergodic hypothesis appears graphically simple, its strict mathematical justification is extremely difficult.

injury

In the case of spontaneous symmetry breaking , the ergodic hypothesis is violated (ergodicity breaking ). There are then disjoint ergodic regions in phase space. This can happen with phase transitions , with glass transitions , ie when a liquid solidifies , or with spin glasses .

Use in systems theory

The term ergodic hypothesis is also used in systems theory to classify systems or the signals they generate : an ergodic signal is a stochastic (i.e. random) stationary signal that is both aperiodic and recurring. This is e.g. This is the case, for example, when the signal has a distinctive waveform without it being repeated at fixed intervals . Ergodic systems tend to produce an output signal that is only slightly dependent on the initial excitation.

literature

Hannelore Bernhardt : About the development and importance of the ergodic hypothesis in the beginnings of statistical physics. NTM 8 (1971) 1, pp. 13-25.

Individual evidence

↑ Statistical Thermodynamics, Normand M. Laurendeau, Cambridge University Press, 2005, ISBN 0521846358 , p. 379, limited preview in the Google book search
↑ on the difference between ergodic and quasi-ergodic and other questions: See Richard Becker : Theory of Heat ("Theory of Heat", 1954). 1st edition. Springer-Verlag, Berlin 1955. p. 97.
↑ on the quantum mechanical justification: see Albert Messiah : Quantenmechanik, Volume 1 ("Mécanique quantique", 1962). 2nd edition DeGruyter, Berlin 1985, p. 17, ISBN 3-11-010265-X .

Web links

Ergodicity in Markov chains

[1] Statistical Thermodynamics, Normand M. Laurendeau, Cambridge University Press, 2005, ISBN 0521846358 , p. 379, limited preview in the Google book search

[2] the difference between ergodic and quasi-ergodic and other questions: See Richard Becker : Theory of Heat ("Theory of Heat", 1954). 1st edition. Springer-Verlag, Berlin 1955. p. 97.

[3] the quantum mechanical justification: see Albert Messiah : Quantenmechanik, Volume 1 ("Mécanique quantique", 1962). 2nd edition DeGruyter, Berlin 1985, p. 17, ISBN 3-11-010265-X .