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]

Contents

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 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.

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 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

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 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:

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In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic behavior of nature from the behavior of such ensembles.

<span class="mw-page-title-main">Thermodynamics</span> Physics of heat, work, and temperature

Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws of thermodynamics which convey a quantitative description using measurable macroscopic physical quantities, but may be explained in terms of microscopic constituents by statistical mechanics. Thermodynamics applies to a wide variety of topics in science and engineering, especially physical chemistry, biochemistry, chemical engineering and mechanical engineering, but also in other complex fields such as meteorology.

In physics, specifically statistical mechanics, an ensemble is an idealization consisting of a large number of virtual copies of a system, considered all at once, each of which represents a possible state that the real system might be in. In other words, a statistical ensemble is a set of systems of particles used in statistical mechanics to describe a single system. The concept of an ensemble was introduced by J. Willard Gibbs in 1902.

<span class="mw-page-title-main">Ideal gas</span> Mathematical model which approximates the behavior of real gases

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<span class="mw-page-title-main">Internal energy</span> Energy contained within a system

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<span class="mw-page-title-main">Thermodynamic system</span> Body of matter in a state of internal equilibrium

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<span class="mw-page-title-main">Non-equilibrium thermodynamics</span> Branch of thermodynamics

Non-equilibrium thermodynamics is a branch of thermodynamics that deals with physical systems that are not in thermodynamic equilibrium but can be described in terms of macroscopic quantities that represent an extrapolation of the variables used to specify the system in thermodynamic equilibrium. Non-equilibrium thermodynamics is concerned with transport processes and with the rates of chemical reactions.

<span class="mw-page-title-main">Ludwig Boltzmann</span> Austrian physicist and philosopher (1844–1906)

Ludwig Eduard Boltzmann was an Austrian physicist and philosopher. His greatest achievements were the development of statistical mechanics, and the statistical explanation of the second law of thermodynamics. In 1877 he provided the current definition of entropy, , where Ω is the number of microstates whose energy equals the system's energy, interpreted as a measure of statistical disorder of a system. Max Planck named the constant kB the Boltzmann constant.

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An ideal Bose gas is a quantum-mechanical phase of matter, analogous to a classical ideal gas. It is composed of bosons, which have an integer value of spin, and abide by Bose–Einstein statistics. The statistical mechanics of bosons were developed by Satyendra Nath Bose for a photon gas, and extended to massive particles by Albert Einstein who realized that an ideal gas of bosons would form a condensate at a low enough temperature, unlike a classical ideal gas. This condensate is known as a Bose–Einstein condensate.

In statistical mechanics, the microcanonical ensemble is a statistical ensemble that represents the possible states of a mechanical system whose total energy is exactly specified. The system is assumed to be isolated in the sense that it cannot exchange energy or particles with its environment, so that the energy of the system does not change with time.

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<span class="mw-page-title-main">Microstate (statistical mechanics)</span> Specific microscopic configuration of a thermodynamic system

In statistical mechanics, a microstate is a specific configuration of a system that describes the precise positions and momenta of all the individual particles or components that make up the system. Each microstate has a certain probability of occurring during the course of the system's thermal fluctuations.

The concept entropy was first developed by German physicist Rudolf Clausius in the mid-nineteenth century as a thermodynamic property that predicts that certain spontaneous processes are irreversible or impossible. In statistical mechanics, entropy is formulated as a statistical property using probability theory. The statistical entropy perspective was introduced in 1870 by Austrian physicist Ludwig Boltzmann, who established a new field of physics that provided the descriptive linkage between the macroscopic observation of nature and the microscopic view based on the rigorous treatment of large ensembles of microstates that constitute thermodynamic systems.

Entropy is one of the few quantities in the physical sciences that require a particular direction for time, sometimes called an arrow of time. As one goes "forward" in time, the second law of thermodynamics says, the entropy of an isolated system can increase, but not decrease. Thus, entropy measurement is a way of distinguishing the past from the future. In thermodynamic systems that are not isolated, local entropy can decrease over time, accompanied by a compensating entropy increase in the surroundings; examples include objects undergoing cooling, living systems, and the formation of typical crystals.

<span class="mw-page-title-main">Boltzmann's entropy formula</span> Equation in statistical mechanics

In statistical mechanics, Boltzmann's equation is a probability equation relating the entropy , also written as , of an ideal gas to the multiplicity, the number of real microstates corresponding to the gas's macrostate:

<span class="mw-page-title-main">Gas</span> State of Matter

Gas is one of the four fundamental states of matter. The others are solid, liquid, and plasma.

<span class="mw-page-title-main">Temperature</span> Physical quantity that expresses hot and cold

Temperature is a physical quantity that expresses quantitatively the attribute of hotness or coldness. Temperature is measured with a thermometer. It reflects the kinetic energy of the vibrating and colliding atoms making up a substance.

<span class="mw-page-title-main">Thermal fluctuations</span> Random temperature-influenced deviations of particles from their average state

In statistical mechanics, thermal fluctuations are random deviations of an atomic system from its average state, that occur in a system at equilibrium. All thermal fluctuations become larger and more frequent as the temperature increases, and likewise they decrease as temperature approaches absolute zero.

References

  1. Hill, Terrell L. (2002). Thermodynamics of Small Systems. Courier Dover Publications. ISBN   9780486495095.
  2. S.J. Blundell and K.M. Blundell, "Concepts in Thermal Physics", Oxford University Press (2009)
  3. 1 2 Huang, Kerson (1987). Statistical Mechanics. Wiley. ISBN   0471815187.