Ehrenfest2023 (1)

Assignment Boltzmann's Dilemma Ehrenfest's Urn Model Paul (1880-1933) and Tatyana (born Afanasjeva, 1905-1984) Ehrenfest came up with another example to illustrate how to reconcile thermodynamics and irreversibility with the underlying reversible laws of classical mechanics. Paul Ehrenfest, born in 1880, was a student of Boltzmann's. The following descrip- tion of Ehrenfest's urn model was taken out of the wonderful book by J.R. Dorfman Introduction to Chaos in Nonequilibrium Statistical Mechanics . Consider two urns, I and II , and a bag. N balls, labeled 1 , 2 · · · ,N , are dis- tributed between the urns. The bag contains N pieces of paper that carry numbers 1 , 2 · · · ,N . At each time step, somebody draws a number out of the bag at random, moves the corresponding ball to the other urn, and then puts the piece of paper back in the bag. This is repeated a large number of times. Consider the (absolute) difference D ( t ) = bardbl N I ( t ) N II ( t ) bardbl between the number of balls in urn I and in urn II as a function of time. (0.) If you can, write a computer programme to simulate this experiment! (1.) Derive an exact equation for the time evolution of the expected value of D ( t ) assuming transition probabilities to move a ball from urn I to urn II , and vice versa. (2.) Make an approximation for the transition probabilities based on macro- scopic variables (your Stoßzahlansatz ) and derive a closed evolution equation for the macroscopic variable of the expected D ( t ). How would you characterize the behaviour of D ( t ) for short and long times? (3.) How can you relate the behaviour of the distribution of the balls in the urns for this model to the resolution of the Zermelo and the Loschmidt paradoxes? What would be the "entropy" for this system with its associated irreversible time evolution? Notice that the chance that a ball will move from the fuller urn to the emptier urn is always greater than the chance that the opposite will happen, therefore the system has the tendency to approach an equilibrium, where the balls are distributed equally between the urns. Fluctuations will also occur, even very large ones. 1
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