1549380323-Statistical Mechanics Theory and Molecular Simulation

Ensemble distribution 69

or ∫

Ω

dxt

[

∂

∂t

f(xt,t) +∇xt·( ̇xtf(xt,t))

]

= 0. (2.5.6)

Since the choice of Ω is arbitrary, eqn. (2.5.6) must hold locally, so that the term in
brackets vanishes identically, giving

∂

∂t

f(xt,t) +∇xt·( ̇xtf(xt,t)) = 0. (2.5.7)

Finally, since∇xt·( ̇xtf(xt,t)) = ̇xt·∇xtf(xt,t) +f(xt,t)∇xt· ̇xt, and the phase space
divergence∇xt· ̇xt= 0, eqn. (2.5.7) reduces to

∂

∂t

f(xt,t) + ̇xt·∇xtf(xt,t) = 0. (2.5.8)

The quantity on the left side of eqn. (2.5.8) is just the total time derivative off(xt,t),
which includes both the time dependence of the phase space vectorxtand the explicit
time dependence off(xt,t). Thus, we obtain finally

df

dt

=

∂

∂t

f(xt,t) + ̇xt·∇xtf(xt,t) = 0, (2.5.9)

which states thatf(xt,t) is conserved along a trajectory. This result is known as the
Liouville equation. The conservation off(xt,t) implies that

f(xt,t) =f(x 0 ,0), (2.5.10)

and since dxt= dx 0 , we have

f(xt,t)dxt=f(x 0 ,0)dx 0. (2.5.11)

Eqn. (2.5.11) states that the fraction of ensemble members in the initial volume element
dx 0 is equal to the fraction of ensemble members in the volume element dxt. Eqn.
(2.5.11) ensures that we can perform averages over the ensembleat any point in time
because the fraction of ensemble members is conserved. Since ̇xt=η(xt,t), eqn. (2.5.9)
can also be written as

df

dt

=

∂

∂t

f(xt,t) +η(xt,t)·∇xtf(xt,t) = 0. (2.5.12)

Writing the Liouville equation this way allows us to recover the “passive” view of the
ensemble distribution function in which we remain at a fixed location in phase space.
In this case, we remove thetlabel attached to the phase space points and obtain the
following partial differential equation forf(x,t):

∂

∂t

f(x,t) +η(x,t)·∇xf(x,t) = 0, (2.5.13)

which is another form of the Liouville equation. Since eqn. (2.5.13) is a partial differ-
ential equation, it can only specify a class of functions as solutions.Specific solutions