scholarly journals Spectral theory of fluctuations in time-average statistical mechanics of reversible and driven systems

2020 ◽  
Vol 2 (4) ◽  
Author(s):  
Alessio Lapolla ◽  
David Hartich ◽  
Aljaž Godec
Keyword(s):  
Statistical Mechanics ◽  
Spectral Theory ◽  
Driven Systems ◽  
Time Average

Phase-space dynamics and ergodicity

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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