Fall 2022 Homework Assignments
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(a) Suppose you have random numbers x uniformly distributed between 0 and 1. Now consider the numbers y=exp(x). (i) How are the numbers y distributed? (ii) What are the average values <y> and <y2> (iii) Construct computer code to generate sufficient number of samples corresponding to the distribution of y and obtain * a histogram for the distribution of these samples * average values corresponding to <y> and <y2>. (b) Consider a harmonic oscillator on the x-axis made up of a mass m and a spring with a restoring force -kx. A random force η(t) with Wiener properties also acts on the mass so that the equations for position and velocity are dx/dt = v dv/dt = -kx/m + η(t)/m Find the Fokker-Planck equation corresponding to the probability distribution function f(x,v,t).
A system has three possible states, say A, B and C. The only possible transitions are from states A to B, B to C, and C to A with equal rates. It is known that the system is in state A at time t=0. (a) Write down the master equation. (b) Find the probabilities for all three states as a function of time.
A complex molecule is made up of K pieces which can be in two states l (long) or s (short). K is very large so that we can consider a single molecule as an ensemble of its parts. The energy of the two states (per part) are El for the l-state and Es for the s-state. (a) Consider the micro-canonical ensemble corresponding to a fixed value of the energy. (Note that a fized value of the energy also corresponds to a fixed length for the molecule.) (i) How many states of the system correspond to this energy? This value corresponds to the volume of the micro-canonical ensemble &Gamma(E); in the Γ space. For the large numbers involved, you can treat the integer variables in these expressions as continuous variables. (ii) Write down the entropy S as a function of the energy. (iii) Obtain a temperature variable from this expression and find the energy and the entropy of the system as a function of the temperature. (b) Now, consider the system in the canonical ensemble and find as function of temperature, (i) the corresponding partition function, (ii) entropy, (iii) and the energy of the system. Discuss the temperature T → 0 and T → ∞ limits for all quantities.
Find the internal energy and light emission spectrum of a one-dimensional "black body" of length L. What is the temperature dependence of maximum emission wavelength and total power emitted?
The scaled Hamiltonian of the strip-Ising model in the figure has the form -βH = K Σi ( SiSi+1 + σiσi+1 + Siσi ) + h Σi ( Si + σi ) with periodic boundary conditions so that SN+1 = S1 and σN+1 = σ1. K and h are the coupling constant and the scaled field. (a) Write down the transfer matrix for the system. Construct computer code to calculate the Helmholtz free energy for the system. Numerically differentiate this quantity to obtain and plot the following quantities: (b) (1/2N) ∂2/∂K2 ln Q ~ Cv for h=0. From the maximum of this curve, identify a "critical coupling" Kc. (c) (1/2N) ∂/∂h ln Q = <S> = <σ> for K = Kc/2 , Kc and 2Kc.
Each homework is to be placed as a single PDF document (which may contain photos of your work) and submitted through moodle.
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