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In physics and physical chemistry, the thermodynamic limit is reached as the number of particles (atoms or molecules) in a system N approaches infinity — or in practical terms, one mole or Avogadro's number ≈ 6 x 1023. The thermodynamic behavior of a system is asymptotically approximated by the results of statistical mechanics as N → ∞, and calculations using the various ensembles converge. Theoretically, this concerns manipulating factorials arising from Boltzmann's formula for the entropy, S = k log W, by using Stirling's approximation, which is justified only when applied to large numbers. But it probably has an empirical basis as well. Ordinary thermodynamics may not apply to collections of only a few atoms or molecules.
In some simple cases, and at thermodynamic equilibrium, the results can be shown to be a consequence of the additivity property of independent random variables; namely that the variance of the sum is equal to the sum of the variances of the independent variables. In these cases, the physics of such systems close to the thermodynamic limit is governed by the central limit theorem in probability.
For systems of large numbers of particles, the genesis of macroscopic behavior from its microscopic origins fades from view. For example, the pressure exerted by a fluid (gas or liquid) is the collective result of collisions between rapidly moving molecules and the walls of a container, and fluctuates on a microscopic temporal and spatial scale. Yet the pressure does not change noticeably on an ordinary macroscopic scale because these variations average out.
Even at the thermodynamic limit, there are still small detectable fluctuations 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 (which 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.
A thermodynamic limit doesn't always exist in all cases. Usually, what people do when taking the thermodynamic limit is to increase the volume together with the particle number so that the average particle number density remains constant. Two common regularizations are the box regularization where matter is confined to a geometrical box, and the periodic regularization where matter is placed in a torus with periodic boundary conditions. But let's look at the following two examples.
