Consider a discrete random variable X with 2n+1 symbols xi, i = 1, 2, …, 2n+1. Determine the upper and lower bounds on the entropy when (a) p(x1)=0 (b) p(x1)=1/2
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Consider a discrete random variable X with 2n+1 symbols xi, i = 1, 2, …, 2n+1. Determine the upper and lower bounds on the entropy when
(a) p(x1)=0
(b) p(x1)=1/2
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- Let p(x, y) be the joint probability distribution of the two random variables X and Y. Define the conditional entropy H(X | Y ) in terms of the joint distribution and associated conditional probabilities.Why is the entropy higher for a system with 5 particles with energy states >1 than for a system with 5 particles with energy states less than or equal to 1?Starting with the Clausius Inequality, ∂S ≥ ∂q/T, can you prove that, under conditions of constant pressure and entropy, for the total entropy to increase, ∂H ≤ 0 J?
- Experimental measurements of the heat capacity of aluminum at low temperatures (below about 50 K) can be fit to the formula Cv = aT+bT3 ,where Cv is the heat capacity of one mole of aluminum, and the constants a and b are approximately a = 0.00135 J/K2 and b = 2.48 X 10-5 J/K4. From this data, find a formula for the entropy of a mole of aluminum as a function of temperature. Evaluate your formula at T = 1 K and at T = 10 K, expressing your answers both in conventional units (J/K) and as unitless numbers (dividing by Boltzmann's constant).Consider a case of n particles and two compartments, if n1 is the number of particles in one compartment and remaining n2=n-n1 particles in other compartment, then the number of microstates in the macrostate (n1, n-n1) or thermodynamic probability isShow that the entropy of a two-state paramagnet, expressed asa function of temperature, is S = Nk [ln(2coshx) -x tanh x], where x = µB/kT.Check that this formula has the expected behavior as T → 0 and T → ∞ .
- In Debye Approximation the entropy at some temperature T (less than 10 K) is aT(Blank 1 ) /3 If the value of this entropy at T = 3.5 K is 1.55 J / K , then the value of the coefficient "a" is : ( Blank 2)Consider a one-dimensional gas consisting of N = 5 particles each of which has the same speed v, but can move in one of two directions with equal probability. The velocity of each particle is independent. What is the probability that all particles are moving in the same direction?Suppose you flip a quarter 10 times. Find the approximate entropy (using Stirling's approximation) divided by the Boltzmann constant ?/?.
- A plastic bag containing 0.2 kg of water at 20°C is dropped from a height of 0.5 m onto an insulating carpet. Assume that the bag does NOT break. What is the approximate probability that a similar bag sitting on a carpet will do the reverse; that is, spontaneously jump 0.5 m in the air? Express your answer in the form "Probability = 10-x," where x is a number you will calculate. (Hint: Note that ey = 10y÷ln(10).)Hi, could I get some help with this macro-connection physics problem involving isothermal expansion? The set up is: For an isothermal reversible expansion of two moles of an ideal gas, what is the entropy change of the a) gas and b) the surroundings in J/K to 4 digits of precision if the gas volume quadruples, assuming NA = 6.022e23 and kB = 1.38e-23 J/K? Thank you.Imagine a photon gas at an initial temperature of T = 1.4 K. What is the temperature of the photon gas (in K) after it has undergone a reversible adiabatic expansion to 2 times its original volume?