Thermodynamic borderline case

The thermodynamic limit case or thermodynamic limit is a central term from statistical physics that establishes the connection between statistical mechanics and thermodynamics . This is the limit behavior of the properties of a system, which is described in the context of statistical physics, when this system is greatly enlarged. Mathematically, the thermodynamic limit is accomplished by performing an asymptotic development .

The thermodynamic limit allows the number of particles and the volume to approach infinity so that the density remains constant: ${\ displaystyle N}$ ${\ displaystyle V}$ ${\ displaystyle N / V}$

{\ displaystyle {\ begin {aligned} & N \ to \ infty, \\ & V \ to \ infty \\ {\ text {mit}} \ quad & N / V = ​​{\ text {const}} \ end {aligned}} }

The most important property of the thermodynamic limit case is in many cases the disappearance of the statistical fluctuations of measured quantities . This makes it possible to speak of a system with thermodynamic state variables (and values for them). Thermodynamics can thus be understood as a thermodynamic borderline case of statistical mechanics.

Example: ideal gas

In the canonical ensemble of classical monatomic ideal gas is subject energy of a single gas atom a random distribution with mean

{\ displaystyle \ langle E_ {1} \ rangle = 3/2 \ cdot k _ {\ mathrm {B}} \ cdot T}

and variance

{\ displaystyle \ langle E_ {1} {} ^ {2} \ rangle - \ langle E_ {1} \ rangle ^ {2} = 3/2 \ cdot \ left (k _ {\ mathrm {B}} \ cdot T \ right) ^ {2}}

With

${\ displaystyle k _ {\ mathrm {B}}}$ for the Boltzmann constant
${\ displaystyle T}$ for the temperature .

Since the atoms of the ideal gas are independent of one another, the mean value and variance of a system of gas atoms result from the central limit theorem as -fold the corresponding value for a particle . ${\ displaystyle N}$ ${\ displaystyle N}$

In the thermodynamic limit value, the relative width of the energy distribution (quotient of standard deviation and expected value ) disappears :

{\ displaystyle \ lim _ {N \ to \ infty} {\ frac {\ sqrt {N \ cdot \ left (\ langle E_ {1} {} ^ {2} \ rangle - \ langle E_ {1} \ rangle ^ {2} \ right)}} {N \ cdot \ langle E_ {1} \ rangle}} = 0}

From this disappearance of the (relative) statistical uncertainty of the energy follows the relation known from the thermodynamics of the ideal gas

{\ displaystyle E = {\ frac {3} {2}} \ cdot N \ cdot k _ {\ mathrm {B}} \ cdot T}

,

in which the total energy of the -particle system is no longer a random variable , but a state variable with a unique value. ${\ displaystyle E}$ ${\ displaystyle N}$

Classification in physics

The thermodynamic borderline case is of fundamental importance within statistical physics, since its existence ensures the applicability of thermodynamics. Outside of statistical physics, the applicability of thermodynamics, and thus implicitly the existence of the thermodynamic limit case, is often simply assumed or has proven to be sufficiently well fulfilled in practice. Despite its important role in statistical physics, the thermodynamic limit case therefore plays practically no role in most areas of physics (or in other sciences).

Phase transitions

In the theory of statistical physics of phase transitions, the following applies: phase transitions only exist in the thermodynamic limit case; finitely large systems cannot have phase transitions. In practice, the behavior of many-particle systems is often so similar to the behavior in the thermodynamic limit case that differences to this are far below the experimental measurement limits. The behavior of such a system is therefore indistinguishable from the boundary behavior. In such cases, despite the finiteness of the system, one speaks of a phase transition.

N-particle computer simulations

In contrast to experimental systems, due to technical limitations (such as storage space and computing time) , computer simulations are often carried out for system variables whose behavior still differs significantly from the thermodynamic limit case. In connection with the computer-based analysis of phase transitions, the problem arises that existing phase transitions may not be recognizable in a simulation. Conversely, the problem arises that the signs for a phase transition seen in a simulation may not exist in the thermodynamic limit case - the phase transition may, for example, be at a different temperature or not exist at all.

In simulations that are not too expensive on the required computing power here, therefore often finite-size scaling used (German about scale finite system size ). In doing so, equivalent systems of different (but still smaller overall) sizes are simulated and then the behavior of the thermodynamic limit value is deduced from the different sizes of the systems.

ensemble

In the thermodynamic limit are ensembles of statistical physics equivalent.

In the micro-canonical ensemble , whose given sizes are the internal energy , the volume and the number of particles , the energy and the number of particles are fixed. In the canonical ensemble , the energy is not fixed, only the temperature , however, applies to the fluctuation of the mean value of the energy ${\ displaystyle E_ {MK}}$ ${\ displaystyle V}$ ${\ displaystyle N}$ ${\ displaystyle T}$

${\ displaystyle {\ frac {\ Delta E} {E_ {K}}} \ propto {\ frac {1} {\ sqrt {N}}}}$ .

In the thermodynamic Limes one can thus define an energy for the canonical ensemble equivalent to the micro-canonical ensemble . The justification for the equivalence of the grand canonical ensemble to the microcanonical and canonical ensemble in the thermodynamic Limes is analogous, whereby the variable number of particles must also be taken into account. ${\ displaystyle E_ {MK} = \ langle E_ {K} \ rangle}$

Sources and individual references

↑ Basic course Theoretical Physics 6: Statistical Physics, Wolfgang Nolting, Springer DE, 2007, p. 373, Google Books
↑ Introduction to Statistical Physics, Kerson Huang, Taylor & Francis, 2001, p. 3, Google Books
↑ These expressions can be calculated from the canonical partition function of the corresponding monatomic gas.
^ Nigel Goldenfeld: Lectures on Phase Transitions and the Renormalization Group. Westview Press, Advanced Book Program, 1992, ISBN 0-201-55409-7 .
↑ G. Orkoulas, Michael Fisher , AZ Panagiotopoulos: Precise simulation of criticality in asymmetric fluids. In: Physical Review. E 63.5 (2001), p. 051507.
↑ Kurt Binder : Finite size scaling analysis of Ising model block distribution functions. In: Journal of Physics. B Condensed Matter 43.2 (1981), pp. 119-140.

[1] Basic course Theoretical Physics 6: Statistical Physics, Wolfgang Nolting, Springer DE, 2007, p. 373, Google Books

[2] Introduction to Statistical Physics, Kerson Huang, Taylor & Francis, 2001, p. 3, Google Books

[3] These expressions can be calculated from the canonical partition function of the corresponding monatomic gas.

[4] Nigel Goldenfeld: Lectures on Phase Transitions and the Renormalization Group. Westview Press, Advanced Book Program, 1992, ISBN 0-201-55409-7 .

[5] G. Orkoulas, Michael Fisher , AZ Panagiotopoulos: Precise simulation of criticality in asymmetric fluids. In: Physical Review. E 63.5 (2001), p. 051507.

[6] Kurt Binder : Finite size scaling analysis of Ising model block distribution functions. In: Journal of Physics. B Condensed Matter 43.2 (1981), pp. 119-140.