Abstract
We discuss the notions of local thermodynamic equilibrium and Le Châtelier’s principle, a corollary of the Second Principle of thermodynamics, as well as some of their consequences. The latter include the positiveness of both specific heat at constant volume and of adiabatic compressibility. As a counterexample, we discuss gravitational collapse from the point of view of thermodynamics. If local thermodynamic equilibrium always holds everywhere, then the general evolution criterion follows. The distinction between continuous and discontinuous systems is introduced, as well as the notion of thermodynamic flux.
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Notes
- 1.
Here, we refer to Sects. 13, 16, 21, 22 and 103 of [1].
- 2.
Here, we assume that this is a global maximum, not a local one. Local maxima of \( S_{TOT} \) correspond to states such that the entropy decreases for an infinitesimal deviation from the state and the body then returns to the initial state, whereas for a finite deviation the entropy may be greater than in the original state. After such a finite deviation the body does not return to its original state, but will tend to pass to some other equilibrium state corresponding to a maximum entropy greater than that in the original state. Accordingly, we must distinguish between ‘metastable’ and ‘stable’ equilibrium states. A body in a metastable state may not return to it after a sufficient deviation. Although a metastable state is stable within certain limits, the body will always leave it sooner or later for another state which is stable, corresponding to the greatest of the possible maxima of entropy. A body which is displaced from this state will always eventually return to it [1]. A problem with local maxima of \( S_{TOT} \) is discussed in Sect. 3.4. See also the note in Sect. 3.2.
- 3.
The actual definitions of x and y are not relevant here. We drop the subscript \( _{TOT} \) in the following for simplicity.
- 4.
Here, we invoke the identity \( \frac{\partial ^2 }{\partial x \partial y} = \frac{\partial ^2 }{\partial y \partial x} \)
- 5.
The quantity (‘Jacobian’) \( \frac{\partial \left( u , v \right) }{\partial \left( a , b \right) } \equiv \frac{\partial u}{\partial a} \frac{\partial v}{\partial b} - \frac{\partial u}{\partial b} \frac{\partial v}{\partial a} \) satisfies the identities \( \frac{\partial \left( u , v \right) }{\partial \left( a , v \right) } = \left( \frac{\partial u}{\partial a} \right) _{v} \) and \( \frac{\partial \left( u , v \right) }{\partial \left( a , b \right) } = \frac{\partial \left( u , v \right) }{\partial \left( c , d \right) } \frac{\partial \left( c , d \right) }{\partial \left( a , b \right) } \) for arbitrary quantities u, v, a, b, c and d [1].
- 6.
It is often impossible to write down rigorous criteria a priori for the validity of LTE in a given problem, and the issue is settled by comparison of LTE-based predictions with observations. This is far from surprising, as similar considerations hold for true thermodynamic equilibrium too—see Sect. 1.5 of Ref. [4].
- 7.
Or, more precisely, that the evolution of the cigarette is a succession of LTE states.
- 8.
Here is why we postulated in Sect. 3.1 that S has only a global maximum, not local maxima. The First Principle of thermodynamics in the form \( dE = TdS - pdV \) gives \( C_v \equiv T ( \frac{\partial S}{\partial T} ) _{v} = ( \frac{\partial E}{\partial T} ) _{v}\) and \( (\frac{dE}{dS})_{v} = T\), hence \( (\frac{dS}{dE})_{v} = \frac{1}{T}\) and \( ( \frac{\partial ^2 S}{\partial E^2} )_{v} = -\frac{1}{T^2}(\frac{dT}{dE})_{v} = - \frac{\left[ (\frac{dS}{dE})_{v} \right] ^2}{C_v} \), i.e. \( C_v = - \frac{\left[ (\frac{dS}{dE})_{v} \right] ^2}{( \frac{\partial ^2 S}{\partial E^2} )_{v}} \) [6]. If \(S = S\left( E \right) \) has local maxima, \( ( \frac{\partial ^2 S}{\partial E^2} )_{v} > 0 \) and \( C_{v} < 0 \) in a neighbourhood of the local minimum of S which is located between two adjacent local maxima of \( S \left( E \right) \), which in turn correspond to metastable states (see note in Sect. 3.1).
- 9.
- 10.
We take into account that \( df = \left( \frac{\partial f}{\partial x} \right) _{y} dx + \left( \frac{\partial f}{\partial y} \right) _{x} dy\) and that \( \frac{\partial ^2 f}{\partial x \partial y} = \frac{\partial ^2 f}{\partial y \partial x}\) for arbitrary differentiable \( f \left( x , y \right) \).
- 11.
Of course, \( \sum _{k} c_{k} = 1 \) identically at all times, hence \( \sum _{k} \dfrac{d c_{k}}{d t} = 0\) at all times.
- 12.
- 13.
- 14.
Here, ‘self-gravitating’ means that each particle is subject only to the Newtonian gravitational attraction of all other particles. As for the contribution to the entropy balance, e.g. of nuclear fusion in stars, see Ref. [10].
- 15.
- 16.
See note of Sect. 3.1.
- 17.
Here, we may refer to the adiabatic compressibility with no loss of generality, so that temperature is not yet involved. Its impact will be discussed below.
- 18.
Realistic models of gravitational collapse include the impact of non-gravitational forces, including the nuclear ones.
- 19.
See the note in Sect. 3.2.
- 20.
For a detailed discussion, see Ref. [6].
- 21.
If we attempt to apply thermodynamics to the entire Universe, regarded as a single isolated system, then [1] we immediately encounter a glaring contradiction [...] any finite region of it, however large, should have a finite relaxation time and should be in equilibrium. Everyday experience shows us, however, that the properties of Nature bear no resemblance to those of an equilibrium system; and astronomical results show that the same is true throughout the vast region of the Universe accessible to our observation. [...] The reason is that, when large regions of the Universe are considered, the gravitational fields [...] may in a sense be regarded as ‘external conditions’ to which the bodies are subject. The statement that an isolated system must, over a sufficiently long time, reach a state of equilibrium applies of course only to a system in steady external conditions. On the other hand, the general cosmological expansion of the Universe means that [...] the ‘external conditions’ are by no means steady [...]. Here, it is important that the gravitational field cannot itself be included in an isolated system: the Universe as a whole must be regarded not as an isolated system but as a system in a variable gravitational field. Consequently, the application of the law of increase of entropy does not prove that [...] equilibrium must necessarily exist.
- 22.
In contrast, it has been shown [14] that a self-consistent thermodynamic description exists for a system (a) which is in thermal equilibrium with a thermal bath; (b) which is made of positive and negative electric charges; (c) where every particle interacts with all other particles via electrostatic, Coulombian interactions; (d) and where the net electric charge is zero everywhere. Every particle undergoes both attractive and repulsive interactions on an equal footing, contrary to the self-gravitating particles which undergo attractive interactions only.
- 23.
The lack of thermal equilibrium between a body A with negative heat capacity and a heat bath B is a particular case, as \( C_{vB} \gg \vert C_{vA} \vert \) by definition of the heat bath. If the specific heat of B is assumed to be large but finite, then a description of the interaction of A and B is available which is in agreement with Le Châtelier’s principle [6].
- 24.
Through the so-called ‘Hawking radiation’, a quantum effect
- 25.
Here, we refer to Sect. 15 of [1].
- 26.
Since \( \sum _{k} c_{k} \equiv 1 \), \( \sum _{k} d c_{k} = d \left( \sum _{k} c_{k} \right) \equiv 0.\)
- 27.
We stress the following two points. Firstly, in steady state \( S = \) constant by definition. Suitable transport processes carry away the amount of entropy produced per unit time at rate \( \frac{dS}{dt} \) in order to maintain \( S = \) const. throughout our system. Secondly, S is often invoked because of the Second Principle of thermodynamics, and the Second Principle does not involve time in its formulation—see note in Sect. 2.2. It is the assumption of LTE everywhere at all times which allows S to be a differentiable function of t, as S is the volume integral of \( \rho s \) and both \( \rho \) and s may depend on both space and time; in particular, \( s = s ( T, p, \ldots )\) just like in thermodynamic equilibrium, and p, T, etc. are solutions of the balance equations of energy, momentum, etc. which depend on space and time.
- 28.
References
Landau, L.D., Lifshitz, E.: Statistical Physics. Pergamon, Oxford, UK (1960)
DeGroot, S.R., Mazur, P.: Non-Equilibrium Thermodynamics. North Holland, Amsterdam (1962)
Landau, L.D., Lifshitz, E.: Fluid Mechanics. Pergamon, Oxford, UK (1960)
Callen, H.B.: Thermodynamics and an Introduction to Thermostatistics, 2nd edn. Wiley, New York (1985)
Fermi, E.: Thermodynamics. Dover Publications (1956)
Velazquez, L.: J. Stat. Mech. Theory Exp. 2016(3), 033105 (2016)
Zemansky, M.W.: Heat and Thermodynamics. McGraw Hill, New York (1968)
Glansdorff, P., Prigogine, I.: Physica 30, 351 (1964)
Hill, T.L.: Thermodynamics of Small Systems. W.A. Benjamin, Inc. Publisher, New York (1964)
Wallace, D.: Br. J. Philos. Sci. 61(3), 513 (2010)
Lynden-Bell, D.: Phys. A 263(1–4), 293–304 (1998)
Padmanabhan, T.: Phys. Rep. 188(5), 285–362 (1990)
Landau, L.D., Lifshitz, E.: Mechanics. Pergamon, Oxford, UK (1960)
Lebowitz, J.L., Lieb, E.H.: PRL 22(13), 631–634 (1969)
Thirring, W.: Z. Physik 235, 339–352 (1970)
LoPresto, M.C.: Phys. Teach. 41, 299–301 (2003)
Shimizu, H., Yamaguchi, Y.: Prog. Theor. Phys. 67, 1 (1982). (Progress Letters)
Di Vita, A.: Phys. Rev. E 81, 041137 (2010)
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Di Vita, A. (2022). Local Thermodynamic Equilibrium. In: Non-equilibrium Thermodynamics. Lecture Notes in Physics, vol 1007. Springer, Cham. https://doi.org/10.1007/978-3-031-12221-7_3
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