1549380323-Statistical Mechanics Theory and Molecular Simulation

Liouville’s theorem 65

point that results from the evolution of x 0. As we noted in Section 1.6, xtis a unique
function of x 0 that can be expressed as xt(x 0 ). Since the mapping of the point x 0 to xt
is one-to-one, this mapping is equivalent to a coordinate transformation on the phase
space from initial phase space coordinates x 0 to phase space coordinates xt. Under
this transformation, the volume element dx 0 transforms according to

dxt=J(xt; x 0 )dx 0 , (2.4.2)

whereJ(xt; x 0 ) is the Jacobian of the transformation, the determinant of the matrix
J defined in eqn. (1.6.28), from x 0 to xt. According to eqn. (1.6.28), the elements of
the matrix are

Jkl=

∂xkt

∂xl 0

. (2.4.3)

We propose to determine the Jacobian in eqn. (2.4.2) by deriving an equation of motion
it obeys and then solving this equation of motion. To accomplish this, we start with
the definition,
J(xt; x 0 ) = det(J), (2.4.4)

analyze the derivative
d
dt
J(xt; x 0 ) =

d

dt

det(J), (2.4.5)

and derive a first-order differential equation obeyed byJ(xt; x 0 ).
The time derivative of the determinant is most easily computed by applying an
identity satisfied by determinants

det(J) = eTr[ln(J)], (2.4.6)

where Tr is the trace operation: Tr(J) =

∑

kJkk. Eqn. (2.4.6) is most easily proved by
first transforming J into a representation in which it is diagonal. If J has eigenvalues
λk, then ln(J) is a diagonal matrix with eigenvalues ln(λk), and the trace operation
yields Tr[ln(J)] =

∑

klnλk. Exponentiating the trace yields

∏

kλk, which is just the
determinant of J. Substituting eqn. (2.4.6) into eqn. (2.4.5) gives

d

dt

J(xt; x 0 ) =

d

dt

eTr[ln(J)]

= eTr[ln(J)]Tr

[

dJ

dt

J−^1

]

=J(xt; x 0 )

∑

k,l

[

dJkl

dt

Jlk−^1

]

. (2.4.7)

The elements of the matrices J−^1 and dJ/dtare easily seen to be

dJkl

dt

=

∂ ̇xkt

∂xl 0

. Jlk−^1 =

∂xl 0

∂xkt

. (2.4.8)

Substituting eqn. (2.4.8) into eqn. (2.4.7) gives