1549380323-Statistical Mechanics Theory and Molecular Simulation

Phase space 7

Another important example of a phase space trajectory is that ofa simple harmonic
oscillator, for which the force law is given by Hooke’s law,F(x) =−kx, wherekis
a constant known as theforce constant. In this case, Newton’s second law takes the
form
m ̈x=−kx. (1.3.5)

For a given initial condition,x(0) andp(0), the solution of eqn. (1.3.5) is

x(t) =x(0) cosωt+

p(0)

mω

sinωt, (1.3.6)

whereω =

√

k/mis the natural frequency of the oscillator. Eqn. (1.3.6) can be
verified by substitution into eqn. (1.3.5). Differentiating once with respect to time and
multiplying by the mass gives an expression for the momentum

p(t) =p(0) cosωt−mωx(0) sinωt. (1.3.7)

Note thatp(t) andx(t) are related by

(p(t))^2

2 m

+

1

2

mω^2 (x(t))^2 =C, (1.3.8)

whereCis a constant determined by the initial condition according to

C=

(p(0))^2

2 m

+

1

2

mω^2 (x(0))^2. (1.3.9)

(This relation is known as theconservation of energy, which we will discuss in greater
detail in the next few sections.) From eqn. (1.3.8), it can be seen that the phase space
plot,pvs.x, specified byp^2 / 2 m+mω^2 x^2 /2 =Cis an ellipse with axes (2mC)^1 /^2 and
(2C/mω^2 )^1 /^2 as shown in Fig. 1.3. The analysis also indicates that different initial

x

p

(2mC)

1/2

(2C/mω 2 )

1/2

Fig. 1.3Phase space of the one-dimensional harmonic oscillator.