Microstate (statistical mechanics)
In contrast, the macrostate of a system refers to its macroscopic properties, such as its temperature, pressure, volume and density.
In this description, microstates appear as different possible ways the system can achieve a particular macrostate.
A macrostate is characterized by a probability distribution of possible states across a certain statistical ensemble of all microstates.
In a quantum system, the microstate is simply the value of the wave function.
All macroscopic thermodynamic properties of a system may be calculated from the partition function that sums
For the more general case of the canonical ensemble, the absolute entropy depends exclusively on the probabilities of the microstates and is defined as where
This form for entropy appears on Ludwig Boltzmann's gravestone in Vienna.
The second law of thermodynamics describes how the entropy of an isolated system changes in time.
The third law of thermodynamics is consistent with this definition, since zero entropy means that the macrostate of the system reduces to a single microstate.
Heat and work can be distinguished if we take the underlying quantum nature of the system into account.
[2] Work is the energy transfer associated with an ordered, macroscopic action on the system.
If this action acts very slowly, then the adiabatic theorem of quantum mechanics implies that this will not cause jumps between energy levels of the system.
The description of a classical system of F degrees of freedom may be stated in terms of a 2F dimensional phase space, whose coordinate axes consist of the F generalized coordinates qi of the system, and its F generalized momenta pi.
But for a system with a huge number of degrees of freedom its exact microstate usually is not important.
So the phase space can be divided into cells of the size h0 = ΔqiΔpi, each treated as a microstate.
Now the microstates are discrete and countable[5] and the internal energy U has no longer an exact value but is between U and U+δU, with
The number of microstates Ω that a closed system can occupy is proportional to its phase space volume:
If the position and momentum of two particles are exchanged, the new state will be represented by a different point in phase space.
possible permutations or possible exchanges of these particles will be counted as part of a single microstate.
For example, in the case of a simple gas of N particles with total energy U contained in a cube of volume V, in which a sample of the gas cannot be distinguished from any other sample by experimental means, a microstate will consist of the above-mentioned N!
points in phase space, and the set of microstates will be constrained to have all position coordinates to lie inside the box, and the momenta to lie on a hyperspherical surface in momentum coordinates of radius U.
If on the other hand, the system consists of a mixture of two different gases, samples of which can be distinguished from each other, say A and B, then the number of microstates is increased, since two points in which an A and B particle are exchanged in phase space are no longer part of the same microstate.
In phase space, the N/2 particles in each box are now restricted to a volume V/2, and their energy restricted to U/2, and the number of points describing a single microstate will change: the phase space description is not the same.
With regard to Boltzmann counting, it is the multiplicity of points in phase space which effectively reduces the number of microstates and renders the entropy extensive.
With regard to Gibbs paradox, the important result is that the increase in the number of microstates (and thus the increase in entropy) resulting from the insertion of the partition is exactly matched by the decrease in the number of microstates (and thus the decrease in entropy) resulting from the reduction in volume available to each particle, yielding a net entropy change of zero.
