Statistical Mechanics

1. Why is internal energy important to statistical mechanics? 2. What is the hypothesis in statistical mechanics in closed system? Internal energy gives the probable energy levels of particles/mole- cules in the system. Microstates of the same energy are equally likely. In other words, all microstates in an isolated system have the same proba- bility. 3. What is a 1-D phase state diagram? A phase state dia- gram is the momen- tum vs. distance dia- gram used to describe microstates. 4. What is the basic assumption of statistical me- chanics? 5. What is the probability distribution function of a microstate? The basic assumption is that we are work- ing with N particles in closed systems. f(x,y,z;p1x, p1y,p1z)dxdydzdpxdpy- dpz 6. What is a microstate of a system specified by? A particular microstate is specified by 6N vari- ables of postiion and momentum. r1,r2,...,rN ; p1,p2,...,pN r1 = (x1, y1, z1) and p = (p1x, p1y,p1z) 7. f(r,p)drdp What is the distribution function used to describe microstates? 8. What is the joint probabilty distribution used to describe microstates in a system? 9. What is the derived equation proving that g = mu in open system from the derivation of GPF? 10. What are two examples of microcanonical ensem- ble? 11. What are the relevance of the lines in the 1-D box and Harmonic oscillator ? By dots 1-D box and Harmonic oscillator where E = 0.5kx^2 + P(x)^2/2m for harmonic oscillator. The microstates on the line have the same en- ergy. 12. What is P(Ek) ? the probability that a system in the mi- crostate k, has energy Ek. 13. How do we specify microstates for discrete lev- els? We specifigy discrete levels with J = 0,1,2... ; For instance J1 = 0 , J2 = 2; For N particle J1...JN = 0. 14. What is ergodicity? (ergodic hypothesis) If we take snap shots of a lot of similar spaces, and trace a particle, the particle will visit all the points over time. It's a way to determine that the frequency of an event iwill remain the same accross a lot of similar systems 15. What is k ? k is the specific mi- crostate. k = f(r1...pN) 16. What is a microstate? A microstates is a very small region in space describe bythe postion and momentum used to determine the probabil- ity of finding a system with energy Ek. 17. What is an ensemble? A collection of mi- crostates 18. What is a micocanonical ensemble? An ensemble of mi- crostates with the same energy 19. What is canonical ensemble? an ensemble of mi- crostates at the same termperature but changing energy. 20. What are two exampels of microcanonical ensem- ble? 21. What is the relevance of the statistical mechanics hypothesis of a closed system? 22. What is the sum of probability of two microstates in a canonical system Particle in 1-D box and Harmonic oscillator Take microstates k1 = a , k2 = b P (Ea+ Eb) = P(Ea).P(Eb) . This is possible because we assume microstates are adiathermic but at constant temperature and do not affect each other. 23. What is P(Ek) P(Ek) = Cexp(-Beta*Ek) 24. How do we derive P(Ek)? 25. What is the partition function? Z = 1/C Z = sum of(Dn*exp(-Beta *Ek)) Z is a normalization constant. 26. What is the probability that the system has energy E? (Number of states with Energy E)*P(E) 27. What is energy level? All states with the same energy. 28. What is the probability that a system is at energy level E (includes all states)? Pl(E) = (#of states of E*P(E))= sum of P(Ek) 29. What is the degeneracy? Degeneracy or weigh- ing factor is the # of states of energy E. 30. How do P(Ek), Pl(E) and #of states of energy E behave as energy increases? 32. What is the expression for dependence of Z on V? 33. What is the expression of Z dependent on beta and volume in closed system? 34. What is the equation for the 1st and 2nd law com- bined and how does it affect internal energy? 35. What are the expressions for dS and P from the internal energy (dU)? 36. What is the equation of state for a perfect monoatomic gas and derivation? 37. What is entropy in terms of K,Z, U and beta? 38. What is Helmholtz free energy F and how is it derived? 39. What is entropy and pressure in terms of Helmholtz free energy and how are they derived? 41. what is entropy in terms of uncertainty and de- rivation? 42. What is P(Ek) at low temperature and high tem- perature? 43. What is fluctuation energy? How is it derived? 44. What is equation of fluctuation in terms of KbT & Cv ? How is it derived? 45. What is the fluctuation of a monoatomic gas? 46. What is the comparison of fluctuation of monoatomic gas and energy of monoatomic gas? 47. What is the energy of fluctuation of a canonical system? DEk^2 = 0 48. What parameters are specified in open systems? Energy (Ek) and Num- ber of Particles (N) 49. What is the second assumption/hypothesis of statistical mechanics for open systems? 50. What is the probability of state of a composite system (A & B)? 51. What is the change of probability of state due to the change in energy and particles? 52. What is the partition function of an open system and how is it derived? 53. What is the grand (canonical ) partition function Z? 54. What is u: energy per particle? 55. What is s; entropy per particle? 56. what is v: volume per particle? 57. What is the total differential of U, S and V? States of the same enr- gy and number of par- ticles have the same probability. Z is The grand partition function. 58. Derive gibbs free energy per particle: 59. Show that g = mu 60. What is p of open system derived from GPF? 61. What is S of open system derived from GPF? 62. What is free energy g of open system derived from GPF? 63. What is the derived equation proving that g = mu in open system from the derivation of GPF? 64. What is the total derivation of the GPF? 65. Evaluate the Gaussian function exp(-x^2 ) `