6 The grand canonical ensemble
6.1 Introduction: The need for yet another ensemble
The ensembles discussed thus far all have the common feature that the particle number
Nis kept fixed as one of the control variables. The fourth ensemble to be discussed,
the grand canonical ensemble, differs in that it permits fluctuationsin the particle
number atconstant chemical potentialμ. Why is such an ensemble necessary? As useful
as the isothermal-isobaric and canonical ensembles are, numerousphysical situations
correspond to a system in which the particle number varies. These include liquid–vapor
equilibria, capillary condensation, and, notably, molecular electronics and batteries, in
which a device is assumed to be coupled to an electron source. In computational
molecular design, one seeks to sample a complete “chemical space” of compounds in
order to optimize a particular property (e.g. binding energy to a target), which requires
varying both the number and chemical identity of the constituent atoms. Finally, in
certain cases, it simply proves easier to work in the grand canonicalensemble, and
given that all ensembles become equivalent in the thermodynamic limit,we are free
to choose the ensemble that proves most convenient for the problem at hand.
In this chapter, we introduce the basic thermodynamics and classical statistical
mechanics of the grand canonical ensemble. We will begin with a discussion of Euler’s
theorem and a derivation of the free energy. Following this, we will consider the par-
tition function of a physical system coupled to both thermal and particle reservoirs.
Finally, we will discuss the procedure for obtaining an equation of state within the
framework of the grand canonical ensemble.
Because of the inherently discrete nature of particle fluctuations, the grand canon-
ical ensemble does not easily fit into the continuous molecular dynamics framework
we have discussed so far for kinetic-energy and volume fluctuations. Therefore, a dis-
cussion of computational approaches to the grand canonical ensemble will be deferred
until Chapters 7 and 8. These chapters will develop the machinery needed to design
computational approaches suitable for the grand canonical ensemble.
6.2 Euler’s theorem
Euler’s theorem is a general statement about a certain class of functions known as
homogeneous functions of degreen. Consider a functionf(x 1 ,...,xN) ofNvariables
that satisfies
f(λx 1 ,...,λxk,xk+1,...,xN) =λnf(x 1 ,...,xk,xk+1,...xN) (6.2.1)