EXAMINATION PAPER

Examination Session: Year: Exam Code: May 2015 MATH4231-WE01

Title: IV

Time Allowed: 3 hours Additional Material provided: None

Materials Permitted: None

Calculators Permitted: Use of electronic calculators is forbidden.

Visiting Students may use dictionaries: Yes

Instructions to Candidates: Credit will be given for: the best TWO answers from Section A, the best THREE answers from Section B, AND the answer to the question in Section C. Questions in Section B and C carry TWICE as many marks as those in Section A.

Revision:

ED01/2015 University of Durham Copyright Page number Exam code 2 of 7 MATH4231-WE01

Useful formulae:

• The volume and surface area of a sphere (of radius R) in n-dimensions are given as πn/2 2 πn/2 Vol(Sn−1) = Rn , Area(Sn−1) = Rn−1 Γ(n/2 + 1) Γ(n/2)

• The one-dimensional Gaussian integral:

∞ r Z 2 π dx e−ax = . −∞ a

• Stirling’s formula: log n! ≈ n log n − n .

• : definition and properties: Z ∞ Γ(x) = e−z zx−1 dz Re(x) > 0 , 0 Γ(x + 1) = x Γ(x) , √ Γ (1/2) = π . • Dirac delta function: Z ∞ dk δ(x) = ei k x −∞ 2π

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SECTION A

1. (a) Derive the Euler relation X U(S,Xi) = TS + Xi Yi i

∂U where Xi are extensive quantities and Yi ≡ are the conjugate intensive ∂Xi quantities describing the system. (b) Write the First Law of in terms of the above quantities. (c) Using the Euler relation and the First Law of Thermodynamics, derive the Gibbs-Duhem relation X S dT + Xi dYi = 0 i 2. Consider a system whose entropic fundamental relation is given as

S(U, V, N) = c U 3 V 2 N1/6 ,

for some positive constant c.

(a) Find the equations of state

T = T (S,V,N) ,P = P (S,V,N)

(b) Find the behaviour of the of the system as T → 0, keeping V and N constant. Does the system satisfy the Third Law of Thermodynamics? (c) Adiabats at fixed N satisfy the relation PV γ = constant. Define an adiabat and find γ.

3. (a) State the Second Law of Thermodynamics. What does this imply about re- versibility of time? (b) Write down the expression for entropy in the and briefly explain its meaning.

(c) Write a more general expression for entropy in terms of probabilities Pr of a given microstate r. Show that this reproduces your answer in the Microcanon- ical ensemble.

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4. Define the following ensembles. In each case, specify which quantities are kept fixed. In each case, write down the density function ρ(qi, pi) in terms of the fixed quantities and the Hamiltonian H(qi, pi).

(a) Microcanonical ensemble EM

(b) EC

(c) EG 5. (a) Starting with the U(S,V,N), define the following thermody- namic potentials via a suitable Legendre transform. (i) F (T,V,N) (ii) H(S,P,N) (iii) Gibbs potential G(T,P,N) (b) Using the form of dU given by the First Law of Thermodynamics, write the differential expressions for each of the above thermodynamics potentials. (c) Use (b) to derive the Maxwell relation

 ∂V   ∂µ  = ∂N S ∂P S

6. Consider a quantum system wherein one has discrete energy levels k labeled by an integer k, each energy level being degenerate with degeneracy gk. Derive the statistical count for the system W {nk} as a function of the occupation numbers {nk} for the case when the particles satisfy (a) Maxwell-Boltzmann (classical) statistics (b) Bose-Einstein statistics (c) Fermi-Dirac statistics

Note that for Bose-Einstein and Fermi-Dirac you should treat the particles as in- distinguishable while for Maxwell-Boltzmann they are distinguishable.

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SECTION B

7. Consider the generalized random walk in 1 dimension with si denoting the displace- ment in the ith step. Define the probability that the ith displacement lies between si and si + dsi to be w(si) dsi. Assume the probability distribution for each step is independent of the previous steps.

(a) Calculate the mean displacement hxi and its variance h(∆x)2i after N steps in terms of the moments of the single-step probability distribution. (b) Show that the probability distribution p(x) for the total displacement x after N steps is given by Z ∞ dk p(x) = e−i k x C(k)N −∞ 2π where C(k) is the Characteristic function for a single step. (c) Consider a 1-dimensional random walk with probability p moving one step to the right and probability 1 − p of moving one step to the left, where each step size is given by a. Write the single-step probability distribution w(s) and using part (b), find the probability distribution p(x) after N steps.

8. Consider N quantum harmonic oscillators in 1 dimension, each with energy spec- trum  1 E = n + ω , n = 0, 1, 2,... n 2 ~ Calculate the thermodynamical properties of the system in the canonical ensemble ensemble as follows:

(a) Write down the partition function for a single oscillator Z1(β), and from this deduce the N-oscillator partition function Z(β, N). (b) Compute the free energy F (T,N). (c) Compute the entropy S(T,N). (d) Compute the internal energy U(T,N). Do the quantum harmonic oscillators obey the ? Justify your answer. (e) Discuss the classical limit of this system. What is the answer to part (d) in the classical limit?

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9. Consider the thermodynamics of a of relativistic particles in d spatial dimensions with Hamiltonian H(q, p) = c |p| (1) in the grand canonical ensemble.

(a) Calculate the grand partition function Ξ(T, V, z) for the system as a function of the T , volume V , and fugacity z. (b) Calculate the corresponding Q-potential Q(T, V, z) for the grand canonical en- semble and use this to calculate the pressure P in terms of the temperature and fugacity. (c) Calculate the average energy and particle number for this system.

10. Consider a quantum system which has a discrete energy spectrum Ek labeled by an integer k. Each energy level is degenerate with degeneracy gk. The particles satisfy , i.e. each state of the system can be occupied by at most p particles.

(a) Write down the number of states of the system W {nk} corresponding to the occupation numbers {nk} at each energy level. (b) If we have N particles and the total energy of the system is E use the result of part (a) to figure out the most probable occupation numbern ˜k of the particles.

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SECTION C

11. (i) Are the following assertions true or false? Justify your answer. (a) are associated with first-order phase transitions. (b) The phases on either side of the critical temperature can have different spatial symmetries. (c) The order parameter is typically an intensive thermodynamic variable. (c) The correlation length ξ → 0 at the critical point. (e) The ground state of any system must necessarily manifest the full symme- try of the Hamiltonian. (f) The Landau approach deals only with macroscopic quantities and is ap- plicable only near the critical point. (ii) Using the fact that in d dimensions the has scaling dimen- sion d + η − 2, find the scaling dimensions of the following quantities: (a) Order-parameter density M/V (b) The quantity H , where H is the conjugate field defined by M = − ∂G . kB T ∂H

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