Thermodynamic limit

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In statistical mechanics, the thermodynamic limit or macroscopic limit,[1] of a system is the limit for a large number N of particles (e.g., atoms or molecules) where the volume V is taken to grow in proportion with the number of particles.[2] The thermodynamic limit is defined as the limit of a system with a large volume, with the particle density held fixed.[3]

In this limit, macroscopic thermodynamics is valid. There, thermal fluctuations in global quantities are negligible, and all thermodynamic quantities, such as pressure and energy, are simply functions of the thermodynamic variables, such as temperature and density. For example, for a large volume of gas, the fluctuations of the total internal energy are negligible and can be ignored, and the average internal energy can be predicted from knowledge of the pressure and temperature of the gas.

Note that not all types of thermal fluctuations disappear in the thermodynamic limit—only the fluctuations in system variables cease to be important. There will still be detectable fluctuations (typically at microscopic scales) in some physically observable quantities, such as

Mathematically an asymptotic analysis is performed when considering the thermodynamic limit.

Origin

The thermodynamic limit is essentially a consequence of the

Avogadro number
of molecules, fluctuations are negligible, and so thermodynamics works. In general, almost all macroscopic volumes of gases, liquids and solids can be treated as being in the thermodynamic limit.

For small microscopic systems, different statistical ensembles (microcanonical, canonical, grand canonical) permit different behaviours. For example, in the canonical ensemble the number of particles inside the system is held fixed, whereas particle number can fluctuate in the grand canonical ensemble. In the thermodynamic limit, these global fluctuations cease to be important.[3]

It is at the thermodynamic limit that the additivity property of macroscopic

six vertex model
: the bulk free energy is different for periodic boundary conditions and for domain wall boundary conditions.

Inapplicability

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 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 on the surface of a flat torus (i.e. box 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 that (unlike the
    gravitational
    systems, where matter tends to clump into filaments, galactic superclusters, galaxies, stellar clusters and stars.
  • A system with a nonzero average 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.
  • Certain
    superfluidity.[citation needed
    ]
  • Any system that is not
    H-stable
    ; this case is also called catastrophic.

References

  1. .
  2. ^ S.J. Blundell and K.M. Blundell, "Concepts in Thermal Physics", Oxford University Press (2009)
  3. ^ .