1549380323-Statistical Mechanics Theory and Molecular Simulation

Problems 73

T

S

Fig. 2.5Thermodynamic cycle.

a. Find a solution of the Liouville equation that also satisfies this initial

distribution.

Hint: Show that the substitutionf(x,t) = eαtf ̃(x,t) yields an equation

for a conserved distributionf ̃(x,t). Next, try multiplying thex^2 in the

initial distribution by an arbitrary functiong(t) that must satisfyg(0) =

1. Use the Liouville equation to derive an equation thatg(t) must satisfy
   and then solve this equation.

b. Describe the evolution of the ensemble distribution qualitatively and ex-

plain why it should evolve this way.

c. Show that your solution is properly normalized in the sense that

∫∞

−∞

dxf(x,t) = 1.

∗2.6. An alternative definition of entropy was proposed by Gibbs, whoexpressed

the entropy in terms of the phase space distribution functionf(x,t) as

S(t) =−k

∫

dxf(x,t) lnf(x,t).