- Thermodynamic limit
In

physics andphysical chemistry , the**thermodynamic limit**is reached as the number of particles (atom s ormolecule s) in a system "N" approaches infinity — or in practical terms, one mole orAvogadro's number ≈ 6 x 10^{23}. The thermodynamic behavior of a system is asymptotically approximated by the results ofstatistical mechanics as "N" → ∞, and calculations using the various ensembles converge. Theoretically, this concerns manipulatingfactorial s arising from Boltzmann's formula for theentropy , "S" = "k" log "W", by usingStirling'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 independentrandom variable s; namely that thevariance 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 thecentral 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 afluid (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 andJohnson-Nyquist noise can be measured.Certain quantum mechanical phenomena near the

absolute zero "T" = 0 present anomalies; e.g., Bose-Einstein condensation,superconductivity andsuperfluid ity.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 andvolume ) is the sum of the two separate values.**Cases where there is no thermodynamic limit**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.

* Particles with an attractive potential which doesn't turn around and become repulsive even at very short distances. In such a case, matter tends to clump together instead of spreading out evenly over all the available space. This is the case for gravitational systems, where matter tends to clump into filaments, galactic superclusters, galaxies, stellar clusturs and stars. Ultimately, it might be favorable for everything to coalesce into a black hole.

* A system with a nonzero charge density also poses the same problem. In this case, periodic boundary conditions are out of question because there's no consistent value for the electric flux. With a box regularization, on the other hand, matter tends to accumulate along the boundary of the box instead of being spread more or less evenly with only minor fringe effects.

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