1. Master Equation
START WITH PHYSICAL PICTURE:
q= microscopic state of dynamic system


Changes by stochastic transitions


Rates W(q_new|q_old) for $q_{old}\Rightarrow q_{new}$ specify model.
World of Fermi's golden rule. In general microscopic fluctuations on timescales
short compared to times of interest. Describe system by P(q,t)
probability of state q at time t.
MASTER EQUATION

• System changes by transitions in and out of states.
• Master equation describes evolution of the probability distribution
  -not evolution of the state of the system.
• Assume system intrinsically stochastic.

SOME SPECIAL CASES
If state is integer n master equation discrete.

$$\frac{\partial P(n,t)}{\partial t}=\sum_{n'}[W(n\vert n')P(n',t)-W(n'\vert n)P(n,t)]$$

One step processes
Birth and death processes can be modeled as taking place one at a time.

$$W(n\vert n')=d(n')\delta_{n,n'-1}+b(n')\delta_{n,n'+1}$$

d=death rate.
b=birth rate.
Natural boundary
For birth and death processes: d(0)=0.
STEADY STATE

$$\frac{\partial P_s(n,t)}{\partial t}=0$$

For all n.
For one step processes must have





Express P(n) in terms of P(0).

$$P_s(n)=\frac{b(n-1)b(n-2)\cdots b(0)}{d(n)d(n-1)\cdots c(1)}P_s(0)$$

Determine P(0) from normalization.

$$\sum_n P_s(n)=1$$

Boxed equation akin to detailed balance. THERMODYNAMIC EQUILIBRIUM:
If steady state also equilibrium (Click here for tutorial on the Boltzmann equilibrium concept),
require:



U(n)= internal energy state with n particles.
w_n degeneracy factor (number of states with n individuals).
F=U-TS= free energy.
$S=k\ln w=$ entropy.
No restriction to one step processes! We will later discuss detailed balance for microscopic processes. Example:
INTRINSIC SEMICONDUCTOR
n=# of electrons=# of holes.
$\Omega=$system size parameter = volume.
$b=\beta\Omega=$electron (hole) generation rate.
$\beta=$constant independent of volume.
$d(n)=\gamma n^2/\Omega=$ recombination rate.
$\gamma=$ another constant independent of $\Omega$.
n=0=natural boundary, i.e. d(0)=0.
Macroscopic rate equation



Expect mean number of carriers in steady state

$$n_0=\Omega\sqrt{\beta/\gamma}$$

Master equation for steady state

d(n)P_s(n)-bP_s(n-1)=0

Substitute $b=\Omega\beta;\;\;d(n)=\gamma n^2/\Omega .$ Express P(1), P(2) etc. in terms of P_s(0).



Solve for P_s(0) using the normalization condition

$$\sum_n P_s(n)=1$$

Use Stirling's formula

$$\ln(n!) = {{1}\over{2}}\ln (2\pi )+(n+{{1}\over{2}})\ln n -n +o(n)$$

find that P_s(n) approximately Gaussian.
Mean and variance

$$\langle n\rangle=n_0=\Omega\sqrt{\beta/\gamma};\;\;\langle (n-n_0)^2\rangle=\frac{n_0}{2}$$

At equilibrium



$\epsilon=$ band gap, $\lambda=$ thermal wavelength

$$\frac{1}{\lambda_{h,e}^3}=\int\frac{d^3p}{h^3} \exp(-\frac{p^2}{2m_{h,e}^*kT})$$

h= Planck const., k Boltzmann const.
m^*= effective mass, T temperature.
Boltzmann/Gibbs formula

$$P_s(n)=\frac{\Omega^{2n}}{Z(n!)^2\lambda_h^{3n}\lambda_e^{3n}}e^{-n\epsilon/kT}$$

Z= normalizing factor (partition function).
$\frac{\Omega^{2n}}{\lambda_h^{3n}\lambda_e^{3n}}=$ phase space factor.
(n!)²= Gibbs factor for identical particles.
$e^{-n\epsilon/kT}=$ Boltzmann factor.
SUMMARY

• Master equation natural starting point when processes intrinsically stochastic.
• Equation describes time evolution of probability distribution.
• When states discrete
  
  
  
  
• Found exact steady state solution for one step processes with a natural boundary.
• Rates restricted if required that steady state agrees with equilibrium Boltzmann-Gibbs statistical physics.
• Mean agrees with rate equation result for large system.

Click here for 2. System size expansion Click here for Return to title page