Thermodynamic limit

Extracted Attributes

• Field of Study

  Statistical mechanics, Thermodynamics, Physics

• Alternative Name

  Macroscopic limit

• Mathematical Tool

  Asymptotic analysis, Central Limit Theorem

• Key Characteristics

  Global thermal fluctuations become negligible; extensive properties become additive; macroscopic thermodynamic relations hold

• Microscopic Exceptions

  Rayleigh scattering, Brownian motion, Johnson–Nyquist noise, Blackbody radiation

• Mathematical Definition

  N → ∞, V → ∞, with N/V = constant

• Application in Computing

  Theoretical efficiency benchmark for non-von Neumann architectures and neuromorphic AI hardware

Timeline

• Publication of 'The Mathematical Theory of Thermodynamic Limits: Thomas-Fermi Type Models' detailing mathematical modeling of macroscopic states. (Source: Oxford Academic / Web Search)

  1998-01-01

Thermodynamic limit

In statistical mechanics, the thermodynamic limit or macroscopic limit, 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. The thermodynamic limit is defined as the limit of a system with a large volume, with the particle density held fixed: N → ∞ , V → ∞ , N V = constant {\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.

Web Search Results

• Thermodynamic limit - Wikipedia

  "macroscopic limit" redirects here. For the limit between microscopic and macroscopic, see microscopic scale and macroscopic scale. For the limit between macroscopic and microscopic celestial bodies, see macroscopic bodies. For the limit between quantum and classical realms, see Quantum mechanics. In statistical mechanics, the thermodynamic limit or macroscopic limit, of a system is the limit "Limit (mathematics)") 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. The thermodynamic limit is defined as the limit of a system with a large volume, with the particle density held fixed: : {\displaystyle N\to \infty ,\,V\to \infty ,\,{\frac {N}{V}}={\text{constant}}} [...] ## Origin [edit] The thermodynamic limit is essentially a consequence of the central limit theorem of probability theory. The internal energy of a gas of N molecules is the sum of order N contributions, each of which is approximately independent, and so the central limit theorem predicts that the ratio of the size of the fluctuations to the mean is of order 1/N1/2. Thus for a macroscopic volume with perhaps 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. [...] 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 and volume) is the sum of the two separate values. In some models of statistical mechanics, the thermodynamic limit exists, but depends on boundary conditions. For example, this happens in six vertex model: the bulk free energy is different for periodic boundary conditions and for domain wall boundary conditions. ## Inapplicability [edit]

• Contents 1 The thermodynamic limit

  the thermodynamic limit is defined as the limit of a system with a large volume V , with the particle density n and energy per particle e that are held fixed:

• What is thermodynamic limit?

  The thermodynamic limit, or macroscopic limit, of a system in statistical mechanics is the limit for a large number N of particles where the

• The Mathematical Theory of Thermodynamic Limits: Thomas ...

  by I Catto · 1998 · Cited by 160 — The thermodynamic limit is a mathematical technique for modeling crystals or other macroscopic objects by considering them as infinite periodic

• Thermodynamic Limit: Limitations & Principles - Vaia

  Thermodynamic Limit allows for the application of the laws of large numbers, which leads to definite and deterministic outcomes from underlying