Consider an extreme relativistic gas consisting of N monatomic N molecules moving in a space of volume V. The Hamiltonian of the system is given by H = c sigma_i=1^N P_i where c denotes the speed of light, and p_i = |p_i| represents the magnitude of the i-th particle's momentum p_i. Note that the gas particles are indistinguishable. (a) Assume that the total energy E is fixed: Compute the number of accessible states sigma(E, V, N) with the constraint such that sigma_i=1^N p_i lessthanorequalto E/c. Show that the entropy of the system is given by S(E, V, N) N k_B [ln(alpha V E^3/N^4)+4] Identify the constant alpha. Derive the equations of state E = E(T, V, N) and P = P(T, V, N). Determine the specific heats C_V and C_P. Find the density of state g(E) from sigma(E, V, N). b) Assume that the system is in equilibrium at some temperature T: Given the dispersion relation epsilon = pc, find out the number of states per unit volume (epsilon) for a particle within the energy range (epsilon, epsilon + d epsilon). Recall that the single-particle density of state g(epsilon) d epsilon = V (epsilon) d epsilon. Compute the partition function Q_N(V, T). Determine the mean energy of the system U. Show that P = 1/3 u where u = U/V Find the relative root-mean-square energy fluctuation Squareroot /U