Phase-space dynamics and ergodicity

Statistical Mechanics: Entropy, Order Parameters, and Complexity ◽

10.1093/oso/9780198865247.003.0004 ◽

2021 ◽

pp. 79-98

Author(s):

James P. Sethna

Keyword(s):

Phase Space ◽

Statistical Mechanics ◽

Dissipative Systems ◽

Planetary Motion ◽

Equilibrium Statistical Mechanics ◽

Average Behavior ◽

Time Average ◽

Space Dynamics ◽

Mathematical Justification ◽

Liouville’S Theorem

This chapter provides the mathematical justification for the theory of equilibrium statistical mechanics. A Hamiltonian system which is ergodic is shown to have time-average behavior equal to the average behavior in the energy shell. Liouville’s theorem is used to justify the use of phase-space volume in taking this average. Exercises explore the breakdown of ergodicity in planetary motion and in dissipative systems, the application of Liouville’s theorem by Crooks and Jarzynski to non-equilibrium statistical mechanics, and generalizations of statistical mechanics to chaotic systems and to two-dimensional turbulence and Jupiter’s great red spot.

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Liouville's theorem and quantum mechanics - time quantization and reality

HNPS Proceedings ◽

10.12681/hnps.2798 ◽

2020 ◽

Vol 9 ◽

pp. 395

Author(s):

C. Syros ◽

G. S. Ioannidis ◽

G. Raptis

Keyword(s):

Distribution Function ◽

Quantum Mechanics ◽

Quantum Theory ◽

Phase Space ◽

Statistical Mechanics ◽

Liouville Equation ◽

Time Variable ◽

Planck Constant ◽

Quantum Numbers ◽

Liouville’S Theorem

The chrono-topology, as introduced axiomatically in a different context, is also supported by Liouville's theorem of statistical mechanics. It is shown that, if time is quantized, the distribution function (d.f.) becomes real. An elementary solution, g, of the classical Liouville equation has been found in phase-space and time, which can be used to construct any differentiable d.f, F(g), satisfying the same Liouville equation. The conditions imposed on F(g) are reality and additivity. The reality requirement, {Im F(g)=0) quantizes: (i) F(g) and makes it time-independent, (ii). The time variable, (iii) The energy. As a verification of chronotopology, the Planck constant h has been calculated on the basis of the time quantization. The d.f. F(g) becomes, after the time quantization, a real generalized Maxwell-Boltzmann d.f, F(g) = exp[g(p, g; l1,l2,..,lN)], depending on Ν quantum numbers. These facts are significant for quantum theory, because they uncover an intrinsic relationship between Liouville's theorem and quantum mechanics.

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The evaluation of Gibbs' phase-integral for imperfect gases

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100011191 ◽

1927 ◽

Vol 23 (6) ◽

pp. 685-697 ◽

Cited By ~ 178

Author(s):

H. D. Ursell

Keyword(s):

Phase Space ◽

Statistical Mechanics ◽

Equations Of Motion ◽

A Priori ◽

Classical Mechanics ◽

Canonical Coordinates ◽

Total Phase ◽

Phase Integral ◽

Liouville’S Theorem ◽

Proper Energy

Statistical mechanics is concerned primarily with what are known as “normal properties” of assemblies. The underlying idea is that of the generalised phase-space. The configuration of an assembly is determined (on classical mechanics) by a certain number of pairs of Hamiltonian canonical coordinates p, q, which are the coordinates of the phase-space referred to. Liouville's theorem leads us to take the element of volume dτ=Πdp dq as giving the correct element of a priori probability. Any isolated assembly is confined to a surface in the phase-space, for its energy at least is constant; when there are no other uniform integrals of the equations of motion, the actual probability of a given aggregate of states of the proper energy, i.e., of a given portion of the surface, varies as the volume, in the neighbourhood of points of this portion, included between two neighbouring surfaces of constant energies E, E + dE; it therefore varies as the integral of (∂E/∂n)−1 taken over the portion. If I be the measure of the total phase-space available, interpreted in this way, and i that of the portion in which some particular condition is satisfied, then i/I is the probability of that condition being satisfied.

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