1549380323-Statistical Mechanics Theory and Molecular Simulation

68 Theoretical foundations

nˆ

dS

Fig. 2.4An arbitrary volume in phase space. dSis a hypersurface element andnˆis the unit
vector normal to the surface at the location of dS.

Fraction of ensemble members in Ω =

∫

Ω

dxtf(xt,t). (2.5.2)

Thus, the rate of decrease of ensemble members in Ω is related to the rate of decrease
of this fraction by

−

d

dt

∫

Ω

dxtf(xt,t) =−

∫

Ω

dxt

∂

∂t

f(xt,t). (2.5.3)

On the other hand, the rate at which ensemble members leave Ω through the surface
can be calculated from the flux, which is the number of ensemble members per unit
area per unit time passing through the surface. Letnˆbe the unit vector normal to the
surface at the point xt(see Fig. 2.4). Then, as a fraction of ensemble members, this
flux is given by ̇xt·nˆf(xt,t). The dot product withnˆensures that we count only those
ensemble members actually leaving Ω through the surface, that is, members whose
trajectories have a component of their phase space velocity ̇xtnormal to the surface.
Thus, the rate at which ensemble members leave Ω through the surface is obtained by
integrating overS:
∫

S

dS ̇xt·ˆnf(xt,t) =

∫

Ω

dxt∇xt·( ̇xtf(xt,t)). (2.5.4)

The right side of eqn. (2.5.6) follows from the divergence theorem applied to the
hypersurface integral. Equating the right sides of eqns. (2.5.4) and (2.5.3) gives
∫

Ω

dxt∇xt·( ̇xtf(xt,t)) =−

∫

Ω

dxt

∂

∂t

f(xt,t), (2.5.5)