4 Classical mechanics
̈r= 0. (1.2.8)The straight line motion of eqn. (1.2.7) is, in fact, the unique solution of eqn. (1.2.8)
for an object whose initial position isr(0) and whose initial (and constant) velocity is
v. Thus, Newton’s second law embodies Newton’s first law.
Statistical mechanics is concerned with the behavior of large numbers of objects
that can be viewed as the fundamental constituents of a particular microscopic model
of the system, whether they are individual atoms or molecules, or even groups of atoms
in a macromolecule (for example, the amino acids in a protein). We shall,henceforth,
refer to these constituents as “particles” (or, in some cases, “pseudoparticles”). The
classical behavior of a system ofNparticles in three dimensions is given by the gener-
alization of Newton’s second law to the system. In order to develop the general form
of Newton’s second law, note that particlei,i∈[1,N],will experience a forceFidue
to all of the other particles in the system and possibly the externalenvironment or ex-
ternal agents as well. Denoting the position vectors of theNparticles asr 1 ,...,rN, the
forcesFiare generally functions of these positions, and if frictional forcesare present,
Ficould also be a function of the particle’s velocityr ̇i. We denote this functional
dependence asFi=Fi(r 1 ,...,rN,r ̇i). For example, if the forceFidepends only on
individual contributions from every other particle in the system, wesay that the forces
arepairwise additive. In this case, the forceFican be expressed as
Fi(r 1 ,...,rN,r ̇i) =∑
j 6 =ifij(ri−rj) +f(ext)(ri,r ̇i). (1.2.9)The first term in eqn. (1.2.9) describes forces that are intrinsic to the system and are
part of the definition of the mechanical system, while the second term describes forces
that are entirely external to the system. For a generalN-particle system, Newton’s
second law for particleitakes the form
mi ̈ri=Fi(r 1 ,...,rN,r ̇i). (1.2.10)These equations, referred to as theequations of motionof the system, must be solved
subject to a set of initial positions,{r 1 (0),...,rN(0)}, and velocities,{r ̇ 1 (0),...,r ̇N(0)}.
In any realistic system, the interparticle forces are highly nonlinearfunctions of theN
particle positions so that eqns. (1.2.10) possess enormous dynamical complexity, and
obtaining an analytical solution is hopeless. Moreover, even if an accurate numerical
solution could be obtained, for macroscopic matter, whereN∼ 1023 , the computa-
tional resources required to calculate and store the solutions foreach and every particle
at a large number of discrete time points would exceed by many orders of magnitude
all those presently available, making such a task equally untenable. Given these con-
siderations, how can we ever expect to calculate physically observable properties of
realistic systems starting from a microscopic description if the fundamental equations
governing the behavior of the system cannot be solved?
The rules of statistical mechanics provide the necessary connection between the
microscopic laws and macroscopic observables. These rules, however, cannot circum-
vent the complexity of the system. Therefore, several approaches can be considered
for dealing with this complexity: A highly simplified model for a system that lends