Thermodynamic limit

The Thermodynamic limit case or Thermodynamic limit is a central concept in statistical physics, which establishes the connection between statistical mechanics and thermodynamics. It involves the limiting behavior of properties of a system described in the context of statistical physics, when this system is greatly enlarged. The thermodynamic limit can be the number of particles N and the volume V tend to infinity so that the density N / V remains constant:

The most important feature of the thermodynamic limiting case is in many cases the disappearance of the statistical fluctuations of measured variables. This allows a system with thermodynamic state variables ( and values ​​for this ) to speak. The thermodynamics can thus be understood as a thermodynamic limiting case of statistical mechanics.

Example: Ideal gas

In the canonical ensemble of a classical monatomic ideal gas is subject to the energy of a single gas atom a random distribution with mean and variance, are being and for the Boltzmann constant and temperature. Since the atoms of an ideal gas are independent of each other, there are the mean and variance of a system of N gas atoms according to the central limit theorem as each N times the Einteilchenmittelwerts and Einteilchenvarianz. In the thermodynamic limit, the relative width of the energy distribution disappears (ratio of standard deviation and expected value ):

For this disappearance of the (relative) statistical uncertainty of the energy which is known from the thermodynamics of the ideal gas relation follows

In which the total energy E of the N-particle system is not a random variable but a state variable with a unique value.

Position in the Physics

The Thermodynamic limit case is within the statistical physics of fundamental importance, as its existence ensures the applicability of thermodynamics. Outside of Statistical Physics, the applicability of thermodynamics, and thus implicitly the existence of the thermodynamic limit Falls, often simply assumed or has been found in practice to comply sufficiently well. Despite its important role in the statistical physics of Thermodynamic limit case, therefore, plays virtually no role in most areas of physics (or other sciences ).

Phase transitions

In theory, the statistical physics of phase transitions is considered: phase transitions exist only in the thermodynamic limit case; finally large systems may not have a phase transition. In practice, the behavior of many-body systems is often already so similar to the behavior in the thermodynamic limit case that differences on this are far below the experimental detection limits. The behavior of such a system is thus indistinguishable from the limiting behavior. One speaks therefore in such cases despite finiteness of the system by a phase transition.

N-particle computer simulations

In contrast to experimental systems are computer simulations due to technical limitations (such as memory and processing time ) is often carried out for system sizes of people whose behavior still differs significantly from the thermodynamic limit. Thus, in the context of computer-based analysis of phase transitions the problem that existing phase transitions may not be seen in a simulation. Conversely, there is the problem that in a simulation seen signs of a phase transition in the thermodynamic limit case may have merit - the phase transition for example, can be at a different temperature or do not exist.

Add the required computing power forth not to expensive simulations is therefore often finite -size scaling ( German as " scale finite system sizes " ) used by equivalent systems of different ( but overall still low ) Size can be simulated and then from the differences between the system sizes the behavior of the thermodynamic limit is closed.

Sources and References

• Statistical Physics