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PHAS0061 Advanced Topics in Statistical Mechanics

创建日期：2024-05-06 16:46:52

所属分类：物理Physical

标签： PHAS0061

Advanced Topics in Statistical Mechanics

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PHAS0061

Advanced Topics in Statistical Mechanics
Answer ALL THREE questions.
The numbers in square brackets indicate the provisional
allocation of maximum marks for different parts of each
question.
[Part marks]
1. (a) It can be said that the world is evolving both reversibly and
irreversibly. Explain the meaning behind these ideas and how
they can be compatible with one another. [4]
(b) Explain how the evolution of the world may be characterised
by the growth of Boltzmann entropy. A decrease in Boltzmann
entropy is considered to be possible but improbable. Why? [4]
(c) Discuss how stochastic entropy production may be regarded
as a measure of the apparent irreversibility of the mechanical
behaviour of physical systems. [3]
(d) A physical property of a system is described as ontic when
it exists independent of the knowledge possessed by an ob-
server about the system. Is entropy ontic? Why? [3]
(e) Two point particles of mass m occupy a 3d box of volume V ,
and interact with one another through an infinite square well
potential that requires them to lie no more than a distance a
apart. They are in canonical equilibrium with a heat bath at
temperature T .
Show that the entropy of the system may be written
S = 3k + k ln (V/v ) and give an expression for v . (In order to
simplify the integration limits you may assume that a3 V ). [6]
PHAS0061/2020-21 CONTINUED
1
[Part marks]
2. (a) The evolution of a quantum system coupled to a heat bath may
be described using stochastic differential equations (SDEs) for
variables x and y :
dx = 2(1− x2) dW
dy = −2y dt − 2xy dW .
Derive an SDE for the quantity P = x2 + y2 and show that
dP = 0 when P = 1. [4]
(b) Writing x = P
1
2 cos θ and y = P
1
2 sin θ, show that the SDE for
the quantity θ = tan−1(y/x), for the particular case when P = 1,
is [6]
dθ = − sin 2θdt − 2 sin θdW .
(c) For small θ, show that this reduces to geometric Brownian mo-
tion and by considering ln θ, obtain θ(t) in this approximation.
Comment on the evolution of θ as t →∞. [4]
(d) For slightly different dynamics, the evolution of the quantum
system is governed instead by the SDE
dθ =
(
1− c2 sin 2θ)dt − 2c sin θdW ,
where c is a constant. Write down the Fokker-Planck equa-
tion for the probability density function p(θ, t) for this case, and
derive the probability current j(θ, t). [3]
(e) For small c, show that the stationary probability density func-
tion for the dynamics in part (d) is given by
pst(θ) ∝ 1 + c2g(θ),
and identify the function g(θ). [3]
PHAS0061/2020-21 CONTINUED
2
[Part marks]
3. A reckless professor has built a time machine but hasn’t quite mas-
tered its operation. Upon activation the machine will take him one
day into the future with probability 1/3, two days into the future with
probability 1/6, one day into the past with probability 1/3 or two
days into the past with probability 1/6.
(a) Determine the probability distribution for the time shift m ex-
perienced by the professor after activating the machine twice,
and show that the mean and standard deviation of the time
shift are zero and two days, respectively. [4]
(b) The professor’s birthday is four days away (in the future!) and
he is so excited he cannot wait. Calculate the probability that
he can reach his birthday by activating the machine no more
than three times. [2]
(c) Write down a master equation (in the form of a map) that de-
scribes the evolution of the probability Pn(m) that after n acti-
vations of the time machine the professor has experienced a
time shift of m days. [2]
(d) Use the master equation together with initial condition P0(m) =
δm0 to show that the characteristic function
Gn(k ) =
∞∑
m=−∞
Pn(m)eikm
is given by C
(
cos k + 12 cos 2k
)n with C = (2/3)n. [3]
(e) By considering dGn/dk and d2Gn/dk2, evaluated at k = 0,
compute the variance of m for arbitrary n. [4]
(Question continued on next page.)
PHAS0061/2020-21 CONTINUED
3
[Part marks]
(Question continued from previous page.)
(f) A cluster of molecules can change its size by the loss or gain
of single molecules (monomers) to or from its environment.
With regard to the probabilities of these processes, in what
circumstances could we employ the idea that a barrier controls
the rate of nucleation of condensed phase droplets from a gas
phase? For what reason might the barrier be represented by
a thermodynamic potential? [3]
(g) Write down a modified Becker-Do¨ring equation to model the
evolution of populations of clusters in a system if, in addition
to monomer exchange, the loss or gain of molecular dimers to
or from the environment were also possible. Might the idea of
a thermodynamic barrier still apply? [2]

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