Question

The most general formula for the entropy of a system is S = -k_B ?_s P_s ln P_s, where P_s is the probability that the system is in state s. a) Assuming that the total number of microstates accessible to a system is ?, show that the entropy of the system is maximum when all ? states occur with equal probability. b) If, on the other hand, we have an ensemble of systems sharing energy (with mean value ?), show that the entropy is maximum when P_s ? exp(-?E_s), ? being a constant determined by the value of ?. c) Further, if we have an ensemble of systems sharing energy (with mean value ?) and also sharing particles (with mean value ?), show that the entropy is maximum when P_s ? exp(-?N_s - ?E_s), ? and ? being constants determined by the values of ? and ?.

          The most general formula for the entropy of a system is
S = -k_B ?_s P_s ln P_s,
where P_s is the probability that the system is in state s.
a) Assuming that the total number of microstates accessible to a system is ?, show that the entropy of the system is maximum when all ? states occur with equal probability.
b) If, on the other hand, we have an ensemble of systems sharing energy (with mean value ?), show that the entropy is maximum when P_s ? exp(-?E_s), ? being a constant determined by the value of ?.
c) Further, if we have an ensemble of systems sharing energy (with mean value ?) and also sharing particles (with mean value ?), show that the entropy is maximum when P_s ? exp(-?N_s - ?E_s), ? and ? being constants determined by the values of ? and ?.

The most general formula for the entropy of a system is
S = -kB ?s Ps ln Ps,
where Ps is the probability that the system is in state s.
a) Assuming that the total number of microstates accessible to a system is ?, show that the entropy of the system is maximum when all ? states occur with equal probability.
b) If, on the other hand, we have an ensemble of systems sharing energy (with mean value ?), show that the entropy is maximum when Ps ? exp(-?Es), ? being a constant determined by the value of ?.
c) Further, if we have an ensemble of systems sharing energy (with mean value ?) and also sharing particles (with mean value ?), show that the entropy is maximum when Ps ? exp(-?Ns - ?Es), ? and ? being constants determined by the values of ? and ?.

Added by Benjamin M.

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