Categories: Physics, Thermodynamic ensembles, Thermodynamics.

Microcanonical ensemble

The microcanonical or NVE ensemble is a statistical model of a theoretical system with constant internal energy UU, volume VV, and particle count NN.

Consider a box with those properties. We now put an imaginary rigid wall inside the box, thus dividing it into two subsystems AA and BB, which can exchange energy (i.e. heat), but no particles. At any time, AA has energy UAU_A, and BB has UBU_B, so that in total U=UA ⁣+ ⁣UBU = U_A \!+\! U_B.

The particles in each subsystem are in a certain microstate (configuration). For a given UU, there is a certain number cc of possible whole-box microstates with that energy, given by:

c(U)=UAUcA(UA)cB(UUA)\begin{aligned} c(U) = \sum_{U_A \le U} c_A(U_A) \: c_B(U - U_A) \end{aligned}

Where cAc_A and cBc_B are the numbers of subsystem microstates at the given energy levels.

The core assumption of the microcanonical ensemble is that each of these microstates has the same probability 1/c1 / c. Consequently, the probability of finding an energy UAU_A in AA is:

pA(UA)=cA(UA)cB(UUA)c(U)\begin{aligned} p_A(U_A) = \frac{c_A(U_A) \: c_B(U - U_A)}{c(U)} \end{aligned}

If a certain UAU_A has a higher probability, then there are more AA-microstates with that energy, so, statistically, for an ensemble of many boxes, we expect that UAU_A is more common.

The maximum of pAp_A will be the most common in the ensemble. Assuming that we have given the boxes enough time to settle, we go one step further, and refer to this maximum as “equilibrium”. In other words, the subsystem microstates at equilibrium are maxima of pAp_A and pBp_B.

We only need to look at pAp_A. Clearly, a maximum of pAp_A is also a maximum of lnpA\ln p_A:

lnpA(UA)=lncA(UA)+lncB(UUA)lnc(U)\begin{aligned} \ln p_A(U_A) = \ln{c_A(U_A)} + \ln{c_B(U - U_A)} - \ln{c(U)} \end{aligned}

Here, in the quantity lncA\ln{c_A}, we recognize the definition of the entropy SAklncAS_A \equiv k \ln{c_A}, where kk is Boltzmann’s constant. We thus multiply by kk:

klnpA(UA)=SA(UA)+SB(UUA)klnc(U)\begin{aligned} k \ln p_A(U_A) = S_A(U_A) + S_B(U - U_A) - k \ln{c(U)} \end{aligned}

Since entropy is additive over subsystems, the total is S=SA+SBS = S_A + S_B. To reach equilibrium, we are thus maximizing the total entropy, meaning that SS is the thermodynamic potential that corresponds to the microcanonical ensemble.

For our example, maximizing gives the following, more concrete, equilibrium condition:

0=kd(lnpA)dUA=SAUA+SBUA=SAUASBUB\begin{aligned} 0 = k \dv{(\ln{p_A})}{U_A} = \pdv{S_A}{U_A} + \pdv{S_B}{U_A} = \pdv{S_A}{U_A} - \pdv{S_B}{U_B} \end{aligned}

By definition, the energy-derivative of the entropy is the reciprocal temperature 1/T1 / T. In other words, equilibrium is reached when both subsystems are at the same temperature:

1TA=SAUA=SBUB=1TB\begin{aligned} \frac{1}{T_A} = \pdv{S_A}{U_A} = \pdv{S_B}{U_B} = \frac{1}{T_B} \end{aligned}

Recall that our partitioning into AA and BB was arbitrary, meaning that, in fact, the temperature TT must be uniform in the whole box. We get this specific result because heat was the only thing that AA and BB could exchange.

The point is that the most likely state of the box maximizes the total entropy SS. We also would have reached that conclusion if our imaginary wall was permeable and flexible, i.e if it allowed changes in volume VAV_A and particle count NAN_A.


  1. H. Gould, J. Tobochnik, Statistical and thermal physics, 2nd edition, Princeton.