Physics Problem Set 4: First-order Transitions, Radiation Equilibrium, and Entropy, Assignments of Statistical mechanics

Download Physics Problem Set 4: First-order Transitions, Radiation Equilibrium, and Entropy and more Assignments Statistical mechanics in PDF only on Docsity! PHYSICS 715 PROBLEM SET 4 DUE 5/2/05 1. First-order transitions. Prob. 17.3 in Huang. 2. Correlation length. Prob. 17.5 in Huang 3. Equilibrium of Radiation, Ionized Hydrogen, and Neutral Hy- drogen. Let us consider the epoch of the universe (t z lo3 y r s . , T z lo4 K) when all of the exotic matter and antimatter has disappeared. Essentially all of the neutrons are contained in a particles, which, for simplicity, we will ignore. This leaves only electrons, protons, and photons. The electrons and protons attract one another and wish to form hydrogen, but the radiation is still hot enough to keep it largely ionized in the reaction (a) Write two equations for the chemical potentials of these four species. (b) The massive particles are nonrelativistic and nondegenerate (kBT >> E ~ ) at these temperatures and densities. Write an equation that relates the number densities, temperature, and chemical potential for each of the massive particles. Take into account the rest masses, noting that (m, + m, - 7nH)c2 = 13.6 eV. (c) Show that the fractional ionization x is related to these densities by Hint: Consider the population as consisting of H and ionized H. Then the fraction of H that is ionized is using thc ile~t~rality condition. (d) Compute the " decoupling temperature" Tdec by the condition x = 112. Use the facts that from essentially this time to the present day, the background radiation has cooled according to Trad(t) -- l /R( t ) and that the current baryon derisit,y is 7 z ~ o = 3 x 10-~/c7n~. 4. Entropy in a comoving volume. At different times in our discussion, we use the fact that the entropy in a conloving volume V = 47rR3/3 is a const,ant,. Derive this equation using the hasic laws of thermodynarnics.