Copasetic Flow

This installment of “It’s Obvious. Not!” looks at: Book: “Statistical Mechanics” Edition: second Authors: R.K. Pathria Publisher: Elsevier Butterworth Heinemann Page: 14 Section 1.3 follows the derivation laid out in 1.2 , but with a variable volume to arrive at: As in 1.2, a parameter is defined that must be equalized: The above subscript notation indicates that N and E are held constant and that V is equal to its equilibrium value. The basic formula of thermodynamics is then stated as: Immediately after which, it is stated that: But how? First, remembering that N and E are constant, rewrite the basic formula above as: Now, using the expression from section 1.2 relating the micro and macro states: We can write: However, so,

This installment of “It’s Obvious. Not!” looks at: Book: “Statistical Mechanics” Edition: second Authors: R.K. Pathria Publisher: Elsevier Butterworth Heinemann Page: 12 The idea in Pathria is to start with a function for the number of microstates as a function of energy and then maximize it to study the implications of a system in equilibrium, (the maximum number of microstates). Pathria skips a few steps in the differentiation and simplification. There shown below to help me and others along. Have fun! Starting with Maximize with respect to . Keep in mind that is a function of : Use the chain rule of differentiation to expand the second partial derivative: The last derivative term simplifies to -1: So, to maximize we have: Now, consider the differentiation of a function . The chain rule gives: Applying this to our result above, we get: