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TSP Problem Sheet 4
Thermal and Statistical Physics
University of Cambridge
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University of Cambridge Cavendish Laboratory
TSP-2020/21 — Thermal and Statistical Physics (Part II)
Problem sheet 4: Interacting systems, stochastic physics
- Virial coecient and radial distribution function An inter-molecular potential takes the form
(r) = 1 r < a ✏ a < r < 2 a 0 r > 2 a.
Within the virial expansion the radial distribution function is expanded in powers of the density. (a) Sketch the form of the density-independent part of the radial distribution function versus r for k B T ✏ and k B T ⌧ ✏. (b) Evaluate the 2nd virial coecient, B 2 (T ), and the Boyle temperature of the gas. (c) Identify a set of reduced units, v ⇤ 0 and T ⇤ , for which B 2 (T ⇤ )/v ⇤ 0 is independent of a and ✏. Sketch B 2 (T ⇤ )/v 0 ⇤ versus T ⇤ .
- Liquid crystal The order parameter for a fluid of rod shaped molecules is their degree of alignment, Q, with Q = 0 corresponding to a disordered fluid, and Q 6 = 0 corresponding to a nematic liquid crystal. The free energy can be written as
F (Q, T ) = a(T T c )Q 2 bQ 3 + cQ 4 ,
where a, b, c and T c are positive constants. This system shows a first order phase transition, at a temperature T ⇤ , between two states with Q = 0 and Q = Q ⇤ . (a) Calculate Q ⇤ and T ⇤ , using the conditions that the free energies of the two states are equal at the transition and that the free energies are stationary in equilibrium. (b) Calculate the latent heat of the transition.
- Coupled order parameters (i) Suppose the free energy of a system can be written as
F = ↵(T T c )P 2 + bP 4 + cP 6 ,
where c > 0. Show that the system can undergo a first order phase transition at temper- ature T = T c + b 2 / 4 ac if b < 0. (ii) The free energy of a ferroelectric crystal can be written as
F = ↵(T T c )P 2 + bP 4 + cP 6 + D"P 2 + E" 2 ,
where P is the polarisation of the crystal and " is the elastic strain. Show that the crystal will undergo a first order phase transition when D 2 / 4 E > b.
- Particle number fluctuations Show that the fluctuations in particle number, N , at constant temperature, T , and volume, V , are given by
hN 2 i = k B T
✓
@N
@μ
◆
T,V
.
TSP-2020/21 Problem sheet 4 1 Michaelmas Term
University of Cambridge Cavendish Laboratory
- Energy fluctuations For a system of N free electrons the statistical weight,⌦( U ), is proportional to exp[(N U/✏ 0 ) 1 / 2 ], where ✏ 0 is about 10 19 J. Calculate the heat capacity, C, of the system at room tempera- ture. Show that the probability distribution of the energy of the system is approximately Gaussian and find the root mean square fractional energy fluctuation,
p hU 2 i/hU i 2 , for a system with N = 10 23 at room temperature. [Answer: C = 2. 8 ⇥ 10 25 J K 1 per electron;
p hU 2 i/hU i 2 = 4. 5 ⇥ 10 11 .]
- Critical fluctuations Find the mean square fluctuation of magnetisation, hM 2 i, as a function of temperature on both sides of the critical point T c of the ferromagnetic phase transition, which can be described by the Landau free energy expansion
F = a(T T c )M 2 + bM 4.
- Anharmonic oscillator A system is governed by the anharmonic Hamiltonian H(x) = ax 2 + bx 4. This is evaluated by considering a harmonic Hamiltonian H ↵ = ↵x 2 with adjustable ↵, which generates a probability distribution ⇢↵ (x) = Z ↵ 1 exp(↵x 2 ). The auxiliary Hamiltonian H ̃ = H ↵ + hH H ↵ i ↵ , where the average h.. ↵ is evaluated with respect to the probability distribution ⇢↵ (x), has the same average as H itself: hHi ↵ = h H ̃i ↵ Explain why ̃F is an upper bound for F
F F ̃ = F ↵ + hH H ↵ i ↵
and use this to show that ̃F most closely approximates F for ↵ = a + 6bhx 2 i ↵. Calculate and sketch the temperature dependence of ↵ and hx 2 i ↵ for a > 0.
- Brownian motion Derive the Stokes-Einstein relationship for the di↵usion constant of particles of radius R in a fluid of viscosity ⌘
D =
k B T 6 ⇡⌘R.
In 1928 Pospisil observed the Brownian motion of soot particles of radius 0⇥ 10 7 m immersed in a water-glycerine solution of viscosity 2⇥ 10 3 kg m 1 s 1 , at a temperature of 292 K. The observed value of hx 2 i was 3⇥ 10 12 m 2 in a 10-second interval. Use these data to determine an estimate for k B and compare it with the modern value.
- Johnson noise An R C circuit is kept in thermal equilibrium at temperature T. In analogy with a damped oscillator with negligible inertia (figure below), it can be modelled by introducing a Langevin (noise) voltage v(t). The charge q on the capacitor follows the di↵erential equation q C
- R q ̇ = v(t)
Use the classical fluctuation-dissipation theorem to show that h|q(!)| 2 i = 2k B T 1+⌧! C 2 ⌧ 2 , where ⌧ = RC. Hence, obtain an expression for the spectrum of voltage noise h|v(!)| 2 i across the series R-C circuit. Compare your result to the expected Johnson noise 4Rk B T , noting that positive and negative frequencies contribute equally to Johnson noise in a frequency interval [f, f + f ].
TSP-2020/21 Problem sheet 4 2 Michaelmas Term
TSP Problem Sheet 4
Module: Thermal and Statistical Physics
University: University of Cambridge
