Content deleted Content added
m →Definition: grammar |
m →The parameter β: clarify |
||
Line 32:
==The parameter β ==
The role or meaning of the parameter <math>\beta</math> can be understood in a variety of different ways. In classical thermodynamics, it is an [[inverse temperature]]. More generally, one would say that it is the variable that is [[Conjugate variables (thermodynamics)|conjugate]] to some (arbitrary) function <math>H</math> of the random variables <math>X</math>. The word ''conjugate'' here is used in the sense of conjugate [[generalized coordinates]] in [[Lagrangian mechanics]], thus, properly <math>\beta</math> is a [[Lagrange multiplier]]. It is not uncommonly called the [[generalized force]]. All of these concepts have in common the idea that one value is meant to be kept fixed, as others, interconnected in some complicated way, are allowed to vary. In the current case, the value to be kept fixed is
Practically speaking, it can be understood by examining the derivation of the partition function using [[maximum entropy method]]s. Here, the parameter appears as a [[Lagrange multiplier]]; the multiplier is used to guarantee that the [[expected value]] of some quantity is preserved by the distribution of probabilities. Thus, in chemistry problems, the use of just one parameter <math>\beta</math> reflects the fact that there is only one expectation value that must be held constant: this is the energy. For the [[grand canonical ensemble]], there are two Lagrange multipliers: one to hold the energy constant, and another (the [[fugacity]]) to hold the particle count constant. In the general case, there are a set of parameters taking the place of <math>\beta</math>, one for each constraint enforced by the multiplier. Thus, for the general case, one has
| |||