Stochastic Processes

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Exercise. The transition from the limited domain −∞ < τ1 < τ2 < ⋯ < τs < ∞ to the full domain with symmetric Qs is especially convenient, if not indispensable, for generalizing the description to random dots on a plane or in space. Write explicitly the functions Qs for the grand-canonical ensemble of an ideal gas in a fixed volume.

Solution. In the context of the grand canonical ensemble with only one chemical species the functions QN are probability distributions over the state of subsystems of the ensemble of volume V . Such subsystems are characterized by number of particles N and total energy Um. Thus QN(Um) is the probability that a subsystem has N particles and is in energy state Um.

In the grand-canonical ensemble:

Um = TS − PV + Nμ

The probability of a state is determined by its entropy (because entropy measures the number of possible configurations out of the total number of configurations available). In this case,

QN(Um) = e−S∕k

where S is the entropy consistent with the number of particles and the energy of the system.

QN(Um) = e−PV kT −Um kT +Nμ kT

The normalization condition in this case is:

∑ N=0∞∑ m=0∞Q N(Um) = 1