Thermodynamic equilibrium
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In thermodynamics, a thermodynamic system is in thermal equilibrium or thermodynamic equilibrium when its macroscopic observables have ceased to change with time -- for example, an ideal gas whose distribution function has stabilised to a Maxwell-Boltzmann distribution. This allows a single temperature to be attributed to the whole system. (See also chemical equilibrium).
The process that leads to a thermodynamic equilibrium is called thermalisation. An example of this is a system of interacting particles that is left undisturbed by outside influences. By interacting, they will share energy/momentum among themselves and reach a state where the global statistics are unchanging in time.
In thermodynamics, the local state of a system at thermodynamic equilibrium is determined by the values of its intensive parameters (examples of intensive parameters include pressure,temperature etc.).
Local Thermodynamic Equilibrium (LTE)
It is useful to distinguish between global and local thermodynamic equilibrium. Global thermodynamic equilibrium means that the intensive parameters are homogeneous throughout the whole system, while local thermodynamic equilibrium (LTE) means that the intensive parameters are varying in space and time, but are varying so slowly that for any point, one can assume thermodynamic equilibrium in some neighborhood about that point. If the description of the system requires variations in the intensive parameters that are too large, the very assumptions upon which the definitions of these intensive parameters are based will break down, and the system will be in neither global nor local equilibrium. For example, it takes a certain number of collisions for a particle to equilibrate to its surroundings. If the average distance it has moved during these collisions removes it from the neighborhood it is equilibrating to, it will never equilibrate, and there will be no LTE. Temperature is, by definition, proportional to the average internal energy of an equilibrated neighborhood. Since there is no equilibrated neighborhood, the very concept of temperature breaks down, and the temperature becomes undefined.
It is important to note that this local equilibrium applies only to massive particles. In a radiating gas, the photons being emitted and absorbed by the gas need not be in thermodynamic equilibrium with each other or with the massive particles of the gas in order for LTE to exist.
As an example, let us take a glass of water which contains a melting ice cube. LTE will exist in this case. The temperature inside the glass can be defined at any point, but it is colder near the ice cube than far away from it. If we look at the energies of the molecules located near a given point, they will be distributed according to the Maxwell-Boltzmann distribution for a certain temperature. If we look at the energies of the molecules located near another point, they will be distributed according to the Maxwell-Boltzmann distribution for another temperature.
Local thermodynamic equilibrium is not a stable state, unless it is artificially maintained (for example, it could be maintained inside the glass of water by regularly adding ice into it in order to compensate for the melting). Transport phenomena are processes which lead a system from local to global thermodynamic equilibrium. Going back to our example, the diffusion of heat will lead our glass of water toward global thermodynamic equilibrium, a state in which the temperature of the glass is completely homogeneous.
Further reading
- F. Mandl, Statistical Physics, Second Edition, John Wiley & Sons (1988).
