1549380323-Statistical Mechanics Theory and Molecular Simulation

Problems 49

p ̇=−mω^2 x−ζ

p

m

. (1.12.4)

It can be seen that this dynamical system has a non-vanishing compressibility

κ(x,p) =

∂x ̇

∂x

+

∂p ̇

∂p

=−

ζ

m

. (1.12.5)

The fact that the compressibility is negative indicates that the effective “phase space
volume” occupied by the system will, as time increases, shrink and eventually collapse
onto a single point in the phase space (x= 0,p= 0) ast→ ∞under the action of
the damping force. All trajectories regardless of their initial condition will eventually
approach this point ast→∞. Consider an arbitrary volume in phase space and let all
of the points in this volume represent different initial conditions for eqns. (1.12.4). As
these initial conditions evolve in time, the volume they occupy will growever smaller
until, ast→∞, the volume tends toward 0. In complex systems, the evolution of such
a volume of trajectories will typically be less trivial, growing and shrinking in time as
the trajectories evolve. If, in addition, the damped oscillator is driven by a periodic
forcing function, so that the equation of motion reads

m ̈x=−mω^2 x−ζx ̇+F 0 cos Ωt, (1.12.6)

then the oscillator will never be able to achieve the equilibrium situationdescribed
above but rather will achieve what is known as asteady state. The existence of a
steady state can be seen by considering the general solution

x(t) = e−γt[Acosλt+Bsinλt] +

F 0

√

(ω^2 −Ω^2 )^2 + 4γ^2 Ω^2

sin(Ωt+β) (1.12.7)

of eqn. (1.12.6), where

γ=

ζ

2 m

λ=

√

ω^2 −γ^2 β= tan−^1

ω^2 −Ω^2

2 γΩ

, (1.12.8)

andAandBare arbitrary constants set by the choice of initial conditionsx(0) and
x ̇(0). In the long-time limit, the first term decays to zero due to the exp(−γt) prefac-
tor, and only the second term remains. This term constitutes thesteady-statesolution.
Moreover, the amplitude of the steady-state solution can becomelarge when the de-
nominator is a minimum. Considering the functionf(Ω) = (ω^2 −Ω^2 )^2 + 4γ^2 Ω^2 , this
function reaches a minimum when the frequency of the forcing function is chosen to
be Ω =ω

√

1 −γ^2 /(2ω^2 ). Such a frequency is called aresonantfrequency. Resonances
play an important role in classical dynamics when harmonic forces arepresent, a phe-
nomenon that will be explored in greater detail in Chapter 3. Driven systems and
steady states will be discussed in greater detail in Chapter 13.