Thermodynamic Limit

Simply speaking, the thermodynamic limit is the large N limit in statistical mechanics, where N is the number of particles, such as (atoms or molecules), in the system. In this limit, thermodynamics is valid. There, fluctuations are negligible, and thermodynamic quantities such as the pressure and energy do not fluctuate from one second to the next, but are functions of the thermodynamic variables such as the temperature, density, etc. For example, for a large volume of gas the fluctuations of the total internal energy will be negligible and can be ignored, and this average internal energy can be predicted from knowledge of the pressure and temperature of the gas.

The thermodynamic limit is essentially a consequence of the central limit theorem of statistics. The internal energy of a gas of N molecules is the sum of order N contributions, each of which is approximately independent, and so the central limit theorem predicts that the ratio of the size of the fluctuations to the mean is of order 1/N1/2. Thus for a macroscopic volume with perhaps Avogadro's number of molecules, fluctuations are negligible, and so thermodynamics works. This is not just true for gases. With few exceptions, almost all macroscopic volumes of gases, liquids and solids can be treated as being in the thermodynamic limit.

Even at the thermodynamic limit there are detectable fluctuations (typically at small length scales) in physical quantities, but this has a negligible effect on most sensible properties of a system. However, microscopic spatial density fluctuations in a gas scatter light (this effect, known as Rayleigh scattering, is why the sky is blue). These fluctuations become quite large near the critical point in a gas/liquid phase diagram. In electronics, shot noise and Johnson–Nyquist noise can be measured.

Certain quantum mechanical phenomena near the absolute zero T = 0 present anomalies; e.g., Bose–Einstein condensation, superconductivity and superfluidity.

It is at the thermodynamic limit that the additivity property of macroscopic extensive variables is obeyed. That is, the entropy of two systems or objects taken together (in addition to their energy and volume) is the sum of the two separate values. In some models of statistical mechanics, the thermodynamic limit exists, but depends on boundary conditions. For example, this happens in six vertex model: the bulk free energy is different for periodic boundary conditions and for domain wall boundary conditions.