2. Consider a microcanonical system with given number of particles N, inter- nal energy U, and...
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2. Consider a microcanonical system with given number of particles N, inter- nal energy U, and volume V. We divide this system into two subsystems with fixed number of particles N; and volume V for i = 1, 2. The total energy is fixed U = U + U, however, the two subsystems can exchange energy, i.e., U and U = U - U are not fixed. (a) Let W(U) be the probability that the energy of subsystem 1 is in the regime between U and U + dU. Show that W (U) x w(U, V, N)w(U U, V2, N). (b) Denote the most probable value of U by U, i.e. U maximises W(U). Write down the condition that determines U and show that the result is consistent with the zeroth law of thermodynamics. (c) Show that in the thermodynamic limit that W(U) exp 1 1 1 [ (Cv + OV) (U - U)] 2KBT CV.1 Cv.2 and that the standard deviation of this Gaussian distribution scales as ~ N. What do you conclude when N is large? 2. Consider a microcanonical system with given number of particles N, inter- nal energy U, and volume V. We divide this system into two subsystems with fixed number of particles N; and volume V for i = 1, 2. The total energy is fixed U = U + U, however, the two subsystems can exchange energy, i.e., U and U = U - U are not fixed. (a) Let W(U) be the probability that the energy of subsystem 1 is in the regime between U and U + dU. Show that W (U) x w(U, V, N)w(U U, V2, N). (b) Denote the most probable value of U by U, i.e. U maximises W(U). Write down the condition that determines U and show that the result is consistent with the zeroth law of thermodynamics. (c) Show that in the thermodynamic limit that W(U) exp 1 1 1 [ (Cv + OV) (U - U)] 2KBT CV.1 Cv.2 and that the standard deviation of this Gaussian distribution scales as ~ N. What do you conclude when N is large?
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Here is a breakdown of the problem A microcanonical system is a system with fixed number of particles N volume V and total energy U The system is divided into two subsystems with fixed numbers of part... View the full answer
Related Book For
Modern Classical Physics Optics Fluids Plasmas Elasticity Relativity And Statistical Physics
ISBN: 9780691159027
1st Edition
Authors: Kip S. Thorne, Roger D. Blandford
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