1549380323-Statistical Mechanics Theory and Molecular Simulation

496 Classical time-dependent statistical mechanics

transport properties. The coefficient of shear viscosity is an example of atransport
coefficient. Other examples of transport coefficients are the diffusion constant, the
thermal conductivity, the coefficient of bulk viscosity, and the electrical conductivity.
Similarly, to measure a vibrational spectrum, such as an infrared orRaman spectrum,
it is necessary to induce transitions between different vibrational states by subjecting
the system to an external electromagnetic field of a given frequency and measuring the
frequencies at which excitations occur (see Chapter 14). In general, the perturbations
needed to measure such dynamical properties are time-dependent and drive the system
slightly away from equilibrium. Thus, in order to calculate dynamical properties, we
need to develop a statistical mechanical framework for treating systems weakly per-
turbed from the equilibrium state by possibly time-dependent external perturbations.
In this chapter, we will develop the classical theory of such weakly perturbed systems,
and in the next chapter, the corresponding quantum theory will bedeveloped.
To see how the effect of a driving force can change the nature of anensemble in
a simple and familiar example, consider the case of a harmonic oscillatorof massm
and frequencyω. In the absence of any driving forces, the equations of motion for
the oscillator are given by eqns. (1.6.30), and the motion conservesthe Hamiltonian
H(x,p) =p^2 / 2 m+mω^2 x^2 /2. The trajectory traces out the phase space curve shown
in Fig. 1.3, which is simply the ellipseH(x,p) =E. Suppose, now, that the oscillator
is subject to an external driving forceFe(t). Hamilton’s equations of motion now read

x ̇=

p

m

, p ̇=−mω^2 x+Fe(t). (13.1.1)

Depending on the form ofFe(t), the phase space trajectory generated by eqns. (13.1.1)
is considerably more complicated than that of the undriven oscillator. For example,
supposeFe(t) =F 0 cos Ωt, with Ω/ω=

√

2. The phase space trajectory generated is
   shown in Fig. 13.2(a). Comparing Fig. 13.2(a) to Fig. 1.3, perhaps themost strik-

x

p

t

x

(a) (b)

Transient

region

Steady-state

region

Fig. 13.2(a) Phase space of a driven oscillator satisfyingmx ̈ = −mω^2 x+F 0 cos Ωt
for Ω/ω =

√

2. (b) Trajectory of a damped-driven oscillator satisfying
   mx ̈=−mω^2 x−γx ̇+F 0 cos Ωtfor Ω/ω=

√

2.