Mid-Term Evaluation for FYS 3130 Spring 2004 The students can chose 1 problem from the Group 1, 1 problem from the Group 2 and 2 problems from the Group 3 (totally 4 problems). The results must be submitted (in a typeset form) on Wednesday, March 24, 2004.

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Group 1

1.1

Simulation of Langevin Dynamics

1. The Langevin equation for a Brownian particle in a fluid may be written as α dP = − P ∆t + dP˜ (t) m

(1)

where α is a drag coefficient and hdP˜ (t)dP˜ (0)i = 2kB T α∆tδt0

(2)

and δt0 = 1 if t²[0, ∆t] and 0 otherwise. Show that by introducing the velocity we may write the above as r α 2kB T α∆t dv = − v∆t + W (t) (3) m m2 where W (t) is an uncorrelated random variable with a Gaussian distribution, zero mean and unit variance, so that hW (t)W (0)i = δt0 . W (t) is sometimes called a Wiener process. 2. Show that by setting m = 1, α = 1 and 2kB T = 1 the equation of motion for the Brownian particle may be written √ dv = −v∆t + W (t) ∆t (4) dx = v∆t (5) and program these equations in your favorite language. You may use the numerical recipe subroutines ran1 and gasdev which deliver random numbers with a flat and Gaussian distribution respectively. 1

3. Plot log10 (hx2 (t)i) as a function of log10 (t) and show that hx2 (t)i increases first as t2 and then as t. t runs to 1000. Show that the different choices ∆t = 0.05, 0.01 and 0.005 gives roughly the same results. It is helpful to average over a couple of independent runs, say a 100 (ensemble averaging) for each choice of ∆t.

1.2

Consists of two sub-problems, A and B:

A. Sound velocity. 1. When a sound wave passes through a liquid or a gas the period of vibration is short compared to the relaxation time needed for a small element of volume of the fluid to exchange energy with the rest of the fluid through heat flow. Hence compression of such a volume element may be considered adiabatic. By analyzing one-dimensional compression/rarefications of the system of fluid contained in a slab of thickness dx, show that the pressure p(x, t) in the fluid satisfies the wave equation 2 ∂ 2p 2∂ p = u ∂t2 ∂x2

(6)

where the velocity of the sound propagation u is a constant given by u = (ρκs )−1/2 . Here ρ is the equilibrium density of the fluid and κs is the adiabatic compressibility κs = −V −1 (∂V /∂p)s , i.e. its compressibility measured under conditions where the fluid is thermally insulated. 2. Calculate the adiabatic compressibility κs of the ideal gas in terms of its pressure and specific heat ratio γ = cp /cv . 3. Find an expression for the velocity of sound in an ideal gas in terms of γ, its molecular weight µ and T . D How does the sound velocity depend on T at fixed p? How does it depend on p at fixed T ? 4. Calculate the velocity of sound in nitrogen N2 gas at room temperature and pressure p. Take γ = 1.4. B. Thermodynamics of Van der Waals gas. Calculate cP and cV for the Van der Waals gas. Hint: First calculate Helmholtz free energy, then entropy, and then use the definitions of the specific heat. 2

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Group 2

2.1

Doppler shift

A gas of atoms, each of mass m, is maintained in a box at temperature T . The atoms emit light which passes (in the x-direction) through a window in the box and can be observed as a spectral line in a spectroscope. A stationary atom would emit light at the sharply defined frequency ν0 . But because of the Doppler effect the frequency of the light emitted from an atom with horizontal velocity vx is not simply ν0 but rather ³ vx ´ ν = ν0 1 + (7) c where c is the velocity of light. As a result, not all of the light that arrives at the spectroscope is at frequency ν0 , instead it is chracterized by some intensity ditribution I(ν)dν specifying the fraction of light intensity lying in the frequency range ν to ν + dν. Calculate (i) The mean frequency hνi . (ii) The root mean square frequence shift ∆ν 2 = h(ν − hνi)2 i (iii) The relative intensity distribution I(ν) of the light measured in the spectroscope.

2.2

Lorentian distribution for random walk steps

Consider a random walk problem in one dimension and supppose that the probabibility of a single displacement between s and s + ds is given by w(s)ds =

1 b ds . π s2 + b2

(8)

Calculate the probability P (x) that the total displacement x after N steps lies between x and x + dx. Does P (x) become a Gaussian when N becomes large? If not, does this violate the central limit theorem? Explain.

2.3

Curie-Weiss theory of a magnet

The simplest equation for a non-ideal magnet has the form m = tanh[β(Jm + h)]

(9)

where β = 1/kT , J is the effective interaction constant, while h is the magnetic field measured in proper units. (i) Consider the case h = 0 and analyze graphically possible solutions of this equation. Hint: rewrite equation in terms of an auxiliary dimensional variable m ˜ ≡ βJm. Show that a spontaneous magnetization appears at T < Tc = J/k. 3

(ii) Simplify Eq. (9) at h = 0 near the critical temperature and analyze the spontaneous magnetization as a function of temperature. Hint: Put T = Tc (1 + τ ) and consider solutions for small m ˜ and τ . (iii) Analyze the magnetization curve m(h) near Tc . Hint: express Eq. (9) in terms of the ˜ ≡ βh ≈ h/J and plot h ˜ as a function of m. variables m ˜ and h ˜ (iv) Plot and analyze the magnetization curves for T /Tc = 0.6 and T /Tc = 1.6.

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Group 3

3.1

Maxwell relations for magnets

Derive Maxwell’s relations for a simple magnetic system. Hint: Analyze the energy conservation law for a magnetic system, then express thermodynamic potential through the proper variables.

3.2

Calculations with probabilities

Three people shoot the same target, partial probabilities to hit the target being p1 , p2 , and p3 , respectively. 1. Find the probability that no one will hit the target. 2. Find the probability to find at least one bullet in the target. 3. P Find the distribution function Pn to find n bullets in the target and check that n Pn = 1. 4. Find average value of the number of bullets in the target, n ¯ , and mean square devi2 2 ation, (∆n) ≡ (n − n ¯) . 5. Find numerical values of Pn , n ¯ , and (∆n)2 for p1 = 0.8, p2 = 0.9, p3 = 0.7.

3.3

Symmetric one dimensional random walk

Consider a asymmetric random walk with the probability p for a hop to the right and q = 1 − p for a hop to the left. The probability WN (m) for for m hops to the right from total number of hops N is given by the binomial distribution WN (m) =

N! pm (1 − p)N −m . m!(N − m)!

1. Find the probability PN (M ) for a total displacement M after N hops. 4

2. Find this probability for a symmetric case, p = 1/2. ¯ after N hops. Hint: it is easier to use the equality 3. Calculate average displacement M ¯ = 2m M ¯ − N and calculate m. ¯ ¢ ¡ ¯ )2 . 4. For the same situation calculate the dispersion (∆M )2 = M − M q ¯ . What happens for symmetric random walk? 5. Compare ∆∗ M ≡ (∆M )2 and M 6. Two drunks start out together at the origin, each having equal probability of making a step to the left or to the right along the x axis. Find the probability that they meet again after each one makes N steps. It is understood that they make steps simultaneously. Hint: It is practical to consider their relative motion.

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