MCP 1 Chapter 2 Statistical Mechanics

1. Not sensitive to precise mis details

2. Averages A (so-called equilibrium ensembles)

3. Connected to a mis function

1. A=A(r^N,q^N) with coordinates r and momenta q

2. Probability density p=p(r^N,q^N)

3. <A>ens = int int dr^N dq^N A(r^N,q^N)*p(r^N,q^N)

1. Time average equals ensemble average over long times

2. <A>ens = <A>time for t→inf

3. Study via molecular dynamics simulation

1. <A>=Sum_i Ai*pi with population p_i

2. p_i = n_i/N = e^(-\beta*eps_i)/q

1. q = Sum_i e^(-\beta*eps_i)

2. q = 1/(1-e^(-\beta*eps)) for equally energy levels

3. q = Sum_l g_l*e^(-\beta*eps_l) for degeneracy g_l

4. lim_T→0 q = g_0 // One surviving term

5. lim_T→inf q = number of molecular states

1. Micro-canonical (N,V,E) 2. canonical (N,V,T) 3. grand-canonical and 4. isobaric-isothermal (N,P,T)

1. Fix energy

2. Fix composition

3. Equal probability for each possible state

1. Varying energy

2. Fix composition

3. Thermal equilibrium with temperature T

4. Probability depends on energy

1. Varying energy

2. Varying composition

3. Thermal equilibrium with temperature T

4. Chemical equilibrium with chemical potentials

5. Probability depends on energy and composition

1. Composition of N particles

2. Population of states n_0 … n_n // {n_0, …, n_n} is called instantaneous configuration

3. Degeneracy g_i max amount in n_i // State_i is seen as a container with max capacity

4. State of energies eps_0 … eps_n // Total energy E

5. “Number of different ways” or “weight of a configuration” W

“All possible energy distribution are equally probable”

1. Distinguishable

2. Independent

3. Not interacting

1. Dominant configuration

2. n_i/N = e^(-\beta*eps_i) / Sum_i e^(-\beta*eps_i) forall i

3. \beta = 1/kT

1. W=N!/Prod_i n_i!

2. ln W = ln N! - Sum_i ln n_i!

3. Sterling approximation with ln x! = x*lnx-x for << x

4. ln W = N ln N – Sum_i n_i ln n_i

W=N!/Prod_i n_i! * Prod_i g_i^n_i

1. Find maximum of function f under constraints g=k

2. L = f +\lambda*g = 0 with dL=0 and df=\lambda*dg // Only g

3. df=\alpha*dg + \beta*dh // g and h

1. P(states)*Density(states) __A__

2. Most members have close to mean energy

1. For distinguishable independent molecules Q=q^N

2. For indistinguishable independent molecules Q=q^N/N!

3. q → Q

4. eps_i → E_i

1. <A> = 1/QN! Int dp^N int dr^N A(p^N,r^N)*e^(-\beta*E(p^N,r^N))

2. E_internal = (d(A/kT)/d(1(kT))_V,N = <E>

1. Similar to canonical ensemble

2. Varying volume but fix total volume

3. Delta = Sum_i e^(-\beta*P*V_i) * Q = Sum_i e^-beta(E_i + P*V_i)

1. S=k*ln W(N,V,E) for micro-canonical

2. A=-k*T*ln Q(N,V,T) for canonical

3. G=-k*T*ln Delta(N,P,T) for isobaric-isothermal

Partition q

1. Similar to thermodynamic entropy

2. S = k ln W by Boltzmann

3. S = -kN Sum_i p_i*ln p_i

4. S = [U(T)-U(0)]/T + k*N*ln q

1. S = k_B*T*(d ln Q/dT)_V,N + k_B ln Q

2. p = k_B*T*(d ln Q/dV)_T,N

3. E = k_B*T²*(d ln Q/dT)_V,N

4. mye_i= -k_B*T*(d ln Q/dN_i)_T,V_a!=i

5. c_V = 2kT*(dlnQ/dT)_V + kT²*(d²lnQ/dT²)_V

6. H – H_0 = U + PV = kT²*(dlnQ/dT)_V + kTV(dlnQ/dV)_T

7. G – G_0 = H – TS = A + PV = -kTlnQ + kTV(dlnQ/dV)_T

8. G – G_0 = -nRT*ln(q_m/N_A) for monoatomic gas with indistinguishable independent molecules Q=q^N/N!

Sum of 1. translation eps^T 2. rotation eps^R 3. vibration eps^V and 4. electronic contribution eps^E

-| translation, rotation, vibration, electronic contribution |+

q = q^T*q^R*q^V*q^E

q^x = Sum_i exp(-\beta*eps_i^x)

1. eps^T = n²h²/8ma² with n speed a length

2. peta = h²/8ma² // For n=1

3. eps_n^T = (n²-1)*peta

1. q^T = Sum_n exp(-\beta*(n²-1)*peta)

2. q^T = sqrt(2pi*mkT/h²)*a = a/Delta // For 1D container

3. q^T = q_x^T + q_y^T + q_z^T = V/Delta³ with V=abc

eps_J^R = hcB*J(J+1)

1. q^R ~ T/sigma*Theta^R with Theta^R = hcB/k

2. sigma is symmetry number // If T>>Theta^R

1. q^R ~ pi^1/2*T^3/2 / sqrt(sigma*Theta^A*Theta^B*Theta^C)

2. Theta^X = h²/8pi²*X*k

1. Three moments of inertia

2. Three rotational constants