1549380323-Statistical Mechanics Theory and Molecular Simulation

574 Langevin and generalized Langevin equations

equivalent toV(q), the term in brackets represents the interaction between the system
and the bath, and the final term is a pure bath–bath interaction.
Suppose the bath potentialUbathcan be reasonably approximated by an expan-
sion up to second order about a minimum characterized by values ̄q,y ̄ 1 ,..., ̄ynof the
generalized coordinates. The condition forUbathto have a minimum at these values is

∂Ubath

∂qα

∣

∣

∣

∣

{q= ̄q,y= ̄y}

= 0, (15.1.7)

where all coordinates are set equal to their values at the minimum. Performing the
expansion up to second order gives

Ubath(q,y 1 ,...,yn)≈Ubath( ̄q,y ̄ 1 ,...,y ̄n) +

∑

α

∂Ubath

∂qα

∣

∣

∣

∣

{q= ̄q,y= ̄y}

(qα−q ̄α)

+

1

2

∑

α,β

(qα−q ̄α)

[

∂^2 Ubath

∂qα∂qβ

∣

∣

∣

∣

{q= ̄q,y= ̄y}

]

(qβ−q ̄β). (15.1.8)

The second term in eqn. (15.1.8) vanishes by virtue of the condition ineqn. (15.1.7).
The first term is a constant that can be made to vanish by shifting the absolute zero
of the potential (which is, anyway, arbitrary). Thus, the bath potential reduces, in this
approximation, to

Ubath(q,y 1 ,...,yn) =

1

2

n∑+1

α=1

n∑+1

β=1

q ̃αHαβq ̃β, (15.1.9)

whereHαβ=∂^2 Ubath/∂qα∂qβ|q= ̄q,{y= ̄y}and ̃qα=qα− ̄qαare the displacements of
the generalized coordinates from their values at the minimum of the potential. Note
that since we have already identified the purelyq-dependent term in eqn. (15.1.6),
theH 11 arising from the expansion of the bath potential can be taken to bezero or
absorbed into theq-dependent functionV(q). Since our treatment from this point on
will refer to the displacement coordinates, we will drop the tildes andletqαrefer to the
displacement of a coordinate from its value at the minimum. Separating the particular
coordinateqfrom the other coordinates gives a potential of the form

Ubath(q,y 1 ,...,yn) =

∑

α

Cαqyα+

1

2

∑n

α=1

∑n

β=1

yαH ̃αβyβ, (15.1.10)

whereCα =H 1 α =Hα 1 andH ̃αβis the n×nblock ofHαβ coupling only the
coordinatesy 1 ,...,yn. The potential, though quadratic, is still somewhat complicated
because all of the coordinates are coupled through the matrixHαβ. Thus, in order to
simplify the potential, we introduce a linear transformation of the coordinatesy 1 ,...,yn
tox 1 ,...,xnvia

yα=

∑n

β=1

Rαβxβ, (15.1.11)