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1 Statistical Mechanics URL:http://www.iopb.res.in/~somen/Courses Set III: General Problems on Stat Mech 1. Entropy: 5...

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1 Statistical Mechanics

URL:http://www.iopb.res.in/~somen/Courses

Set III: General Problems on Stat Mech 1. Entropy: 50L of hot water at 55C is added with 25L of cold water at 10C. How much entropy is produced by the time equilibrium occurs (numerical value)? Estimate the change in the number of states? (Hint: Which specific heat to use: cP , cV , or anyother? Get the value from the net.) 2. thermodynamics and computers: Suppose you erase a gigabyte of data on your hard disk. How much heat must accompany this process? (Hint: A gigabyte= 233 bits, a classical bit has two states 0, 1. How many possible states for a gigabyte? A clean disk consists of all 0’s.) 3. Temperature: Consider a collection of a large number of noninteracting Spin-1 particles, each with energy states: E = −, 0, +. It is known that the average energy of a particle is E = − 21 ,. Find the effective temperature of the system. If, in addition, it is known that the mean squared energy of a particle is E 2 = is not in equilibrium. [Take  = 1 and kB = 1, if you want.]

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4. Ultra-relativistic gas in one dimension Consider N ultra-relativistic particles in a one-dimensional box of P size L. Energy of a particle of momentum p is  = c|p| with −∞ < p < +∞. Total energy E = i i . (a) Find (i) the free energy per particle f (T, L) at temperature T , (ii) the equation of state (the analog of P V = N kB T ), and (iii) the chemical potential. Note that P here is the tension (actually negative of tension). Keep N !. (b) Find the equation for an adiabat L = L(T, T0 , L0 ) that goes through the point (T0 , L0 ) in the T, L plane. 5. DNA: Consider a molecular zipper with N links. Each link has a state in which it is closed with energy − ( > 0), and a state in which it is open with energy 0. Let us assume that the zipper can unzip only from the right end (left end is anchored), and that link number s can only open if all links to the right (s + 1, s + 2, ..., N ) are already open. (a) Evaluate the partition function of the zipper. (b) Now assume that there are also molecules that can block the zipper. These molecules bind to site M of the zipper with energy −b and prevent sites s ≥ M from closing. Treat the blocking molecules as an ideal gas of Nb  1 particles in a container of volume V . Compute the probability that the zipper is blocked from closing as a function of the binding site position M and the concentration of blocking molecules c = Nb /V.

Blocked Gas Free Figure 1: Molecular zipper. On the right, with a blocking molecule bound to the zipper while on the left, there are no bound molecules. The red link indicates the only spot where the blocking molecule can bind.