1549380323-Statistical Mechanics Theory and Molecular Simulation

System coupled to a bath 573

the system coordinateqis a simple coordinate, such as a distance between two atoms
or a Cartesian spatial direction (in Section 15.7, we will introduce a general framework
for treating the problem that allows this restriction to be lifted). The Hamiltonian for
qand its conjugate momentumpin the absence of the bath can then be written simply
as

H(q,p) =

p^2

2 μ

+V(q), (15.1.1)

whereμis the mass associated withqandV(q) is a potential energy contribution
that depends only onqand, therefore, is present even without the bath. The system is
coupled to the bath via a potentialUbath(q,y 1 ,...,yn) that involves both the coupling
terms between the system and the bath and terms describing the interactions among
the bath degrees of freedom. The total potential is

U(q,y 1 ,...,yn) =V(q) +Ubath(q,y 1 ,...,yn). (15.1.2)

As an example, consider a system originally formulated in Cartesian coordinates
r 1 ,...,rNdescribed by a pair potential

U(r 1 ,...,rN) =

∑N

i=1

∑N

j=i+1

u(|ri−rj|). (15.1.3)

Suppose the distancer=|r 1 −r 2 |between atoms 1 and 2 is a coordinate of interest,
which we take as the system coordinate. All other degrees of freedom are assigned
as bath coordinates. Suppose, further, that atoms 1 and 2 havethe same mass. We
first transform to the center of mass and relative coordinates between atoms 1 and 2
according to

R=

1

2

(r 1 +r 2 ), r=r 1 −r 2 , (15.1.4)

the inverse of which is

r 1 =R+

1

2

r, r 2 =R−

1

2

r. (15.1.5)

The potential can then be expressed as

U(r 1 ,...,rN) =u(|r 1 −r 2 |) +

∑N

i=3

[u(|r 1 −ri|) +u(|r 2 −ri|)] +

∑N

i=3

∑N

j=i+1

u(|ri−rj|)

=u(r) +

∑N

i=3

[

u

(∣

∣

∣

∣R+

1

2

rn−ri

∣

∣

∣

∣

)

+u

(∣

∣

∣

∣R−

1

2

rn−ri

∣

∣

∣

∣

)]

+

∑N

i=3

∑N

j=i+1

u(|ri−rj|), (15.1.6)

wheren= (r 1 −r 2 )/|r 1 −r 2 |=r/ris the unit vector along the relative coordinate
direction. Eqn. (15.1.6) is of the same form as eqn. (15.1.2), in which the first term is