Thermodynamic limit

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]

${\displaystyle N\to \infty ,\,V\to \infty ,\,{\frac {N}{V}}={\text{constant}}}$

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

• microscopic spatial density fluctuations in a gas scatter light (Rayleigh scattering)
• motion of visible particles (Brownian motion)
• electromagnetic field fluctuations, (blackbody radiation in free space, Johnson–Nyquist noise in wires)

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