Thermodynamic limit
Encyclopedia
In thermodynamics
, particularly statistical mechanics
, the thermodynamic limit is reached as the number of particles (atom
s or molecule
s) in a system, N, approaches infinity. The thermodynamic behavior of a system is asymptotically approximated by the results of statistical mechanics as N tends to infinity, and calculations using the various ensembles used in statistical mechanics converge.
The mathematical basis of this result comes from manipulating factorial
s arising from Boltzmann's
formula for the entropy
, S = k log W by using Stirling's approximation
, which is justified only when applied to large numbers. Empirically, the relative size of fluctuations from the average is much bigger from collections of only a few atoms or molecules, and so the probabilistic assumptions of statistical mechanics break down.
In some simple cases, and at thermodynamic equilibrium
, the results can be shown to be a consequence of the additivity property of independent
random variable
s; namely that the variance
of the sum is equal to the sum of the variances of the independent variables. In these cases, the physics of such systems close to the thermodynamic limit is governed by the central limit theorem
in probability.
For systems of large numbers of particles, the microscopic origins of macroscopic behavior fade from view. For example, the pressure
exerted by a fluid
(gas or liquid) is the collective result of collisions between rapidly moving molecules and the walls of a container, and fluctuates on a microscopic temporal
and spatial scale
. Yet the pressure does not change noticeably on an ordinary macroscopic scale because these variations average out.
Even at the thermodynamic limit, there are still small detectable fluctuations in physical quantities, but this has a negligible effect on most sensible properties of a system. However, microscopic spatial density fluctuations in a gas scatter light (this effect, known as Rayleigh scattering
, is why the sky is blue). These fluctuations become quite large near the critical point in a gas/liquid phase diagram
. In electronics, shot noise
and Johnson–Nyquist noise
can be measured.
Certain quantum mechanical
phenomena near the absolute zero
T = 0 present anomalies; e.g., Bose–Einstein condensation
, superconductivity
and superfluid
ity.
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 thermodynamic limit exists, but depends on boundary conditions. For example this happen in six vertex model: the bulk free energy is different for periodic boundary conditions and for domain wall boundary conditions.
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 in a torus with periodic boundary conditions. However, the following two examples demonstrate cases where these approaches do not lead to a thermodynamic limit:
Thermodynamics
Thermodynamics is a physical science that studies the effects on material bodies, and on radiation in regions of space, of transfer of heat and of work done on or by the bodies or radiation...
, particularly statistical mechanics
Statistical mechanics
Statistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...
, the thermodynamic limit is reached as the number of particles (atom
Atom
The atom is a basic unit of matter that consists of a dense central nucleus surrounded by a cloud of negatively charged electrons. The atomic nucleus contains a mix of positively charged protons and electrically neutral neutrons...
s or molecule
Molecule
A molecule is an electrically neutral group of at least two atoms held together by covalent chemical bonds. Molecules are distinguished from ions by their electrical charge...
s) in a system, N, approaches infinity. The thermodynamic behavior of a system is asymptotically approximated by the results of statistical mechanics as N tends to infinity, and calculations using the various ensembles used in statistical mechanics converge.
The mathematical basis of this result comes from manipulating factorial
Factorial
In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n...
s arising from Boltzmann's
Ludwig Boltzmann
Ludwig Eduard Boltzmann was an Austrian physicist famous for his founding contributions in the fields of statistical mechanics and statistical thermodynamics...
formula for the entropy
Entropy
Entropy is a thermodynamic property that can be used to determine the energy available for useful work in a thermodynamic process, such as in energy conversion devices, engines, or machines. Such devices can only be driven by convertible energy, and have a theoretical maximum efficiency when...
, S = k log W by using Stirling's approximation
Stirling's approximation
In mathematics, Stirling's approximation is an approximation for large factorials. It is named after James Stirling.The formula as typically used in applications is\ln n! = n\ln n - n +O\...
, which is justified only when applied to large numbers. Empirically, the relative size of fluctuations from the average is much bigger from collections of only a few atoms or molecules, and so the probabilistic assumptions of statistical mechanics break down.
In some simple cases, and at thermodynamic equilibrium
Thermodynamic equilibrium
In thermodynamics, a thermodynamic system is said to be in thermodynamic equilibrium when it is in thermal equilibrium, mechanical equilibrium, radiative equilibrium, and chemical equilibrium. The word equilibrium means a state of balance...
, the results can be shown to be a consequence of the additivity property of independent
Statistical independence
In probability theory, to say that two events are independent intuitively means that the occurrence of one event makes it neither more nor less probable that the other occurs...
random variable
Random variable
In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...
s; namely that the variance
Variance
In probability theory and statistics, the variance is a measure of how far a set of numbers is spread out. It is one of several descriptors of a probability distribution, describing how far the numbers lie from the mean . In particular, the variance is one of the moments of a distribution...
of the sum is equal to the sum of the variances of the independent variables. In these cases, the physics of such systems close to the thermodynamic limit is governed by the central limit theorem
Central limit theorem
In probability theory, the central limit theorem states conditions under which the mean of a sufficiently large number of independent random variables, each with finite mean and variance, will be approximately normally distributed. The central limit theorem has a number of variants. In its common...
in probability.
For systems of large numbers of particles, the microscopic origins of macroscopic behavior fade from view. For example, the pressure
Pressure
Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...
exerted by a fluid
Fluid
In physics, a fluid is a substance that continually deforms under an applied shear stress. Fluids are a subset of the phases of matter and include liquids, gases, plasmas and, to some extent, plastic solids....
(gas or liquid) is the collective result of collisions between rapidly moving molecules and the walls of a container, and fluctuates on a microscopic temporal
Time
Time is a part of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify rates of change such as the motions of objects....
and spatial scale
Scale (spatial)
Spatial scale provides a "shorthand" form for discussing relative lengths, areas, distances and sizes. A microclimate, for instance, is one which might occur in a mountain valley or near a lakeshore, whereas a megatrend is one which involves the whole planet....
. Yet the pressure does not change noticeably on an ordinary macroscopic scale because these variations average out.
Even at the thermodynamic limit, there are still small detectable fluctuations in physical quantities, but this has a negligible effect on most sensible properties of a system. However, microscopic spatial density fluctuations in a gas scatter light (this effect, known as Rayleigh scattering
Rayleigh scattering
Rayleigh scattering, named after the British physicist Lord Rayleigh, is the elastic scattering of light or other electromagnetic radiation by particles much smaller than the wavelength of the light. The particles may be individual atoms or molecules. It can occur when light travels through...
, is why the sky is blue). These fluctuations become quite large near the critical point in a gas/liquid phase diagram
Phase diagram
A phase diagram in physical chemistry, engineering, mineralogy, and materials science is a type of chart used to show conditions at which thermodynamically distinct phases can occur at equilibrium...
. In electronics, shot noise
Shot noise
Shot noise is a type of electronic noise that may be dominant when the finite number of particles that carry energy is sufficiently small so that uncertainties due to the Poisson distribution, which describes the occurrence of independent random events, are of significance...
and Johnson–Nyquist noise
Johnson–Nyquist noise
Johnson–Nyquist noise is the electronic noise generated by the thermal agitation of the charge carriers inside an electrical conductor at equilibrium, which happens regardless of any applied voltage...
can be measured.
Certain quantum mechanical
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
phenomena near the absolute zero
Absolute zero
Absolute zero is the theoretical temperature at which entropy reaches its minimum value. The laws of thermodynamics state that absolute zero cannot be reached using only thermodynamic means....
T = 0 present anomalies; e.g., Bose–Einstein condensation
Bose–Einstein condensate
A Bose–Einstein condensate is a state of matter of a dilute gas of weakly interacting bosons confined in an external potential and cooled to temperatures very near absolute zero . Under such conditions, a large fraction of the bosons occupy the lowest quantum state of the external potential, at...
, superconductivity
Superconductivity
Superconductivity is a phenomenon of exactly zero electrical resistance occurring in certain materials below a characteristic temperature. It was discovered by Heike Kamerlingh Onnes on April 8, 1911 in Leiden. Like ferromagnetism and atomic spectral lines, superconductivity is a quantum...
and superfluid
Superfluid
Superfluidity is a state of matter in which the matter behaves like a fluid without viscosity and with extremely high thermal conductivity. The substance, which appears to be a normal liquid, will flow without friction past any surface, which allows it to continue to circulate over obstructions and...
ity.
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
Energy
In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...
and volume
Volume
Volume is the quantity of three-dimensional space enclosed by some closed boundary, for example, the space that a substance or shape occupies or contains....
) is the sum of the two separate values. In some models of statistical mechanics thermodynamic limit exists, but depends on boundary conditions. For example this happen in six vertex model: the bulk free energy is different for periodic boundary conditions and for domain wall boundary conditions.
Cases where there is no thermodynamic limit
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 numberParticle number
The particle number of a thermodynamic system, conventionally indicated with the letter N, is the number of constituent particles in that system. The particle number is a fundamental parameter in thermodynamics which is conjugate to the chemical potential. Unlike most physical quantities, particle...
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 in a torus with periodic boundary conditions. However, the following two examples demonstrate cases where these approaches do not lead to a thermodynamic limit:
- Particles with an attractive potential which doesn't turn around and become repulsive even at very short distances: In such a case, matter tends to clump together instead of spreading out evenly over all the available space. This is the case for gravitationGravitationGravitation, or gravity, is a natural phenomenon by which physical bodies attract with a force proportional to their mass. Gravitation is most familiar as the agent that gives weight to objects with mass and causes them to fall to the ground when dropped...
al systems, where matter tends to clump into filaments, galactic superclusters, galaxies, stellar clusters and stars. - A system with a nonzero charge densityCharge densityThe linear, surface, or volume charge density is the amount of electric charge in a line, surface, or volume, respectively. It is measured in coulombs per meter , square meter , or cubic meter , respectively, and represented by the lowercase Greek letter Rho . Since there are positive as well as...
: 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. - Any system that is not H-stable, this case is also called catastrophic.