Statistical Interpretation of Entropy Statistical mechanics bridges between phenomenological thermodynamicsunderlying microscopic behavior -Statistical.

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Statistical Interpretation of Entropy Statistical mechanics bridges between phenomenological thermodynamicsunderlying microscopic behavior -Statistical mechanics requires a separate lecture (see textbook part II) but abstract nature of entropy asks for an intuitive picture for state function S Here: heuristic approach to statistical interpretation of entropy by Ludwig Boltzman

Consider a gas in a box from an atomistic point of view atoms (or molecules) are moving in a disordered manner within the box having collisions with the walls V1V1 V2V2 Probability to find a particular atom in V 2 reads Let's pick two atom: Probability to find both of them at the same time in V 2 reads

Probability to find all N atom at the same time in V 2 reads V1V1 V2V2 Let W 1 be the “thermodynamic probability” to find system in the homogeneously occupied state (more precisely: # of possible microstates) Thermodynamic probability to find system in a state with all atoms in V 2 reads: or Note: W 1 >>W 2 for N large

Entropy S quantifies the thermodynamic probability W of a particular state We know: S extensive SASA SBSB S A+B Thermodynamic probability W A+B to find combined system A+B in a state where subsystem A is in a state of thermodynamic probability W A and subsystem B is in a state of thermodynamic probability W B reads + = Which function does the job so that

Check: = Let’s determine C We know entropy change with volume changeV 0  V f for an ideal gas in an adiabatically isolated box (see lecture)see lecture Now we apply with

W: # of possible microstates comparison with Principle of increase of entropy Adiabatically insulated system approaches state of maximum probability.