Thermodynamic Limit - Cases Where There Is No Thermodynamic Limit

Cases Where There Is No Thermodynamic Limit

A thermodynamic limit does not exist in all cases. Usually, a model is taken to the thermodynamic limit by increasing the volume together with the particle number while keeping the particle number density 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. However, the following three examples demonstrate cases where these approaches do not lead to a thermodynamic limit:

• 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 clusters and stars.
• A system with a nonzero charge density: In this case, periodic boundary conditions cannot be used because there is 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.
• Any system that is not H-stable, this case is also called catastrophic.