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==Scope==

==Scope==

===Difference between equilibrium and non-equilibrium thermodynamics===

===Difference between equilibrium and non-equilibrium thermodynamics===

===Non-equilibrium state variables===

===Non-equilibrium state variables===

==Overview==

==Overview==

Non-equilibrium thermodynamics is a work in progress, not an established edifice. This article is an attempt to sketch some approaches to it and some concepts important for it.

Non-equilibrium thermodynamics is a work in progress, not an established edifice. This article is an attempt to sketch some approaches to it and some concepts important for it.

===Quasi-radiationless non-equilibrium thermodynamics of matter in laboratory conditions===

===Quasi-radiationless non-equilibrium thermodynamics of matter in laboratory conditions===

According to Wildt<ref name="Wildt 1972">{{Cite journal |last=Wildt |first=R. |year=1972 |title=Thermodynamics of the gray atmosphere. IV. Entropy transfer and production |journal=Astrophysical Journal |volume=174 |issue= |pages=69–77 |doi=10.1086/151469 |bibcode=1972ApJ...174...69W}}</ref> (see also Essex<ref name="Essex 1984a">{{Cite journal |last=Essex |first=C. |year=1984a |title=Radiation and the irreversible thermodynamics of climate |journal=Journal of the Atmospheric Sciences |volume=41 |issue=12 |pages=1985–1991 |doi=10.1175/1520-0469(1984)041<1985:RATITO>2.0.CO;2 |bibcode = 1984JAtS...41.1985E |doi-access=free }}.</ref><ref name="Essex 1984b">{{Cite journal |last=Essex |first=C. |year=1984b |title=Minimum entropy production in the steady state and radiative transfer |journal=Astrophysical Journal |volume=285 |issue= |pages=279–293 |doi=10.1086/162504 |bibcode=1984ApJ...285..279E}}</ref><ref name="Essex 1984c">{{Cite journal |last=Essex |first=C. |year=1984c |title=Radiation and the violation of bilinearity in the irreversible thermodynamics of irreversible processes |journal=Planetary and Space Science |volume=32 |pages=1035–1043 |doi=10.1016/0032-0633(84)90060-6 |bibcode = 1984P&SS...32.1035E |issue=8 }}</ref>), current versions of non-equilibrium thermodynamics ignore radiant heat; they can do so because they refer to laboratory quantities of matter under laboratory conditions with temperatures well below those of stars. At laboratory temperatures, in laboratory quantities of matter, thermal radiation is weak and can be practically nearly ignored. But, for example, atmospheric physics is concerned with large amounts of matter, occupying cubic kilometers, that, taken as a whole, are not within the range of laboratory quantities; then thermal radiation cannot be ignored.

According to Wildt<ref name="Wildt 1972">{{Cite journal |last=Wildt |first=R. |year=1972 |title=Thermodynamics of the gray atmosphere. IV. Entropy transfer and production |journal=Astrophysical Journal |volume=174 |issue= |pages=69–77 |doi=10.1086/151469 |bibcode=1972ApJ...174...69W}}</ref> (see also Essex<ref name="Essex 1984a">{{Cite journal |last=Essex |first=C. |year=1984a |title=Radiation and the irreversible thermodynamics of climate |journal=Journal of the Atmospheric Sciences |volume=41 |issue=12 |pages=1985–1991 |doi=10.1175/1520-0469(1984)041<1985:RATITO>2.0.CO;2 |bibcode = 1984JAtS...41.1985E |doi-access=free }}.</ref><ref name="Essex 1984b">{{Cite journal |last=Essex |first=C. |year=1984b |title=Minimum entropy production in the steady state and radiative transfer |journal=Astrophysical Journal |volume=285 |issue= |pages=279–293 |doi=10.1086/162504 |bibcode=1984ApJ...285..279E}}</ref><ref name="Essex 1984c">{{Cite journal |last=Essex |first=C. |year=1984c |title=Radiation and the violation of bilinearity in the irreversible thermodynamics of irreversible processes |journal=Planetary and Space Science |volume=32 |pages=1035–1043 |doi=10.1016/0032-0633(84)90060-6 |bibcode = 1984P&SS...32.1035E |issue=8 }}</ref>), current versions of non-equilibrium thermodynamics ignore radiant heat; they can do so because they refer to laboratory quantities of matter under laboratory conditions with temperatures well below those of stars. At laboratory temperatures, in laboratory quantities of matter, thermal radiation is weak and can be practically nearly ignored. But, for example, atmospheric physics is concerned with large amounts of matter, occupying cubic kilometers, that, taken as a whole, are not within the range of laboratory quantities; then thermal radiation cannot be ignored.

===Local equilibrium thermodynamics===

===Local equilibrium thermodynamics===

The terms 'classical irreversible thermodynamics'<ref name="Lebon Jou Casas-Vázquez 2008"/> and 'local equilibrium thermodynamics' are sometimes used to refer to a version of non-equilibrium thermodynamics that demands certain simplifying assumptions, as follows. The assumptions have the effect of making each very small volume element of the system effectively homogeneous, or well-mixed, or without an effective spatial structure, and without kinetic energy of bulk flow or of diffusive flux. Even within the thought-frame of classical irreversible thermodynamics, care<ref name="Lavenda 1978"/> is needed in choosing the independent variables<ref>Prigogine, I., Defay, R. (1950/1954). ''Chemical Thermodynamics'', Longmans, Green & Co, London, page 1.</ref> for systems. In some writings, it is assumed that the intensive variables of equilibrium thermodynamics are sufficient as the independent variables for the task (such variables are considered to have no 'memory', and do not show hysteresis); in particular, local flow intensive variables are not admitted as independent variables; local flows are considered as dependent on quasi-static local intensive variables.

The terms 'classical irreversible thermodynamics'<ref name="Lebon Jou Casas-Vázquez 2008"/> and 'local equilibrium thermodynamics' are sometimes used to refer to a version of non-equilibrium thermodynamics that demands certain simplifying assumptions, as follows. The assumptions have the effect of making each very small volume element of the system effectively homogeneous, or well-mixed, or without an effective spatial structure, and without kinetic energy of bulk flow or of diffusive flux. Even within the thought-frame of classical irreversible thermodynamics, care<ref name="Lavenda 1978"/> is needed in choosing the independent variables<ref>Prigogine, I., Defay, R. (1950/1954). ''Chemical Thermodynamics'', Longmans, Green & Co, London, page 1.</ref> for systems. In some writings, it is assumed that the intensive variables of equilibrium thermodynamics are sufficient as the independent variables for the task (such variables are considered to have no 'memory', and do not show hysteresis); in particular, local flow intensive variables are not admitted as independent variables; local flows are considered as dependent on quasi-static local intensive variables.

====Local equilibrium thermodynamics with materials with "memory"====

====Local equilibrium thermodynamics with materials with "memory"====

A further extension of local equilibrium thermodynamics is to allow that materials may have "memory", so that their [[constitutive equation]]s depend not only on present values but also on past values of local equilibrium variables. Thus time comes into the picture more deeply than for time-dependent local equilibrium thermodynamics with memoryless materials, but fluxes are not independent variables of state.<ref>{{cite journal | last1 = Coleman | first1 = B.D. | last2 = Noll | first2 = W. | year = 1963 | title = The thermodynamics of elastic materials with heat conduction and viscosity | url = | journal = Arch. Ration. Mach. Analysis | volume = 13 | issue = 1| pages = 167–178 | doi=10.1007/bf01262690| bibcode = 1963ArRMA..13..167C }}</ref>

A further extension of local equilibrium thermodynamics is to allow that materials may have "memory", so that their [[constitutive equation]]s depend not only on present values but also on past values of local equilibrium variables. Thus time comes into the picture more deeply than for time-dependent local equilibrium thermodynamics with memoryless materials, but fluxes are not independent variables of state.<ref>{{cite journal | last1 = Coleman | first1 = B.D. | last2 = Noll | first2 = W. | year = 1963 | title = The thermodynamics of elastic materials with heat conduction and viscosity | url = | journal = Arch. Ration. Mach. Analysis | volume = 13 | issue = 1| pages = 167–178 | doi=10.1007/bf01262690| bibcode = 1963ArRMA..13..167C }}</ref>

===Extended irreversible thermodynamics===

===Extended irreversible thermodynamics===

'''Extended irreversible thermodynamics''' is a branch of non-equilibrium thermodynamics that goes outside the restriction to the local equilibrium hypothesis. The space of state variables is enlarged by including the [[flux]]es of mass, momentum and energy and eventually higher order fluxes.

'''Extended irreversible thermodynamics''' is a branch of non-equilibrium thermodynamics that goes outside the restriction to the local equilibrium hypothesis. The space of state variables is enlarged by including the [[flux]]es of mass, momentum and energy and eventually higher order fluxes.

==Basic concepts==

==Basic concepts==

There are many examples of stationary non-equilibrium systems, some very simple, like a system confined between two thermostats at different temperatures or the ordinary [[Couette flow]], a fluid enclosed between two flat walls moving in opposite directions and defining non-equilibrium conditions at the walls. [[Laser]] action is also a non-equilibrium process, but it depends on departure from local thermodynamic equilibrium and is thus beyond the scope of classical irreversible thermodynamics; here a strong temperature difference is maintained between two molecular degrees of freedom (with molecular laser, vibrational and rotational molecular motion), the requirement for two component 'temperatures' in the one small region of space, precluding local thermodynamic equilibrium, which demands that only one temperature be needed. Damping of acoustic perturbations or shock waves are non-stationary non-equilibrium processes. Driven [[complex fluids]], turbulent systems and glasses are other examples of non-equilibrium systems.

There are many examples of stationary non-equilibrium systems, some very simple, like a system confined between two thermostats at different temperatures or the ordinary [[Couette flow]], a fluid enclosed between two flat walls moving in opposite directions and defining non-equilibrium conditions at the walls. [[Laser]] action is also a non-equilibrium process, but it depends on departure from local thermodynamic equilibrium and is thus beyond the scope of classical irreversible thermodynamics; here a strong temperature difference is maintained between two molecular degrees of freedom (with molecular laser, vibrational and rotational molecular motion), the requirement for two component 'temperatures' in the one small region of space, precluding local thermodynamic equilibrium, which demands that only one temperature be needed. Damping of acoustic perturbations or shock waves are non-stationary non-equilibrium processes. Driven [[complex fluids]], turbulent systems and glasses are other examples of non-equilibrium systems.

==Stationary states, fluctuations, and stability==

==Stationary states, fluctuations, and stability==

==Local thermodynamic equilibrium==

==Local thermodynamic equilibrium==

The scope of present-day non-equilibrium thermodynamics does not cover all physical processes. A condition for the validity of many studies in non-equilibrium thermodynamics of matter is that they deal with what is known as ''local thermodynamic equilibrium''.

The scope of present-day non-equilibrium thermodynamics does not cover all physical processes. A condition for the validity of many studies in non-equilibrium thermodynamics of matter is that they deal with what is known as ''local thermodynamic equilibrium''.

===Ponderable matter===

===Ponderable matter===

''Local thermodynamic equilibrium of matter''<ref name="Gyarmati 1970"/><ref name="G&P 1971"/><ref name="Balescu 1975"/><ref name="Mihalas Mihalas 1984"/><ref name="Schloegl 1989"/> (see also Keizer (1987)<ref name="Keizer 1987"/> means that conceptually, for study and analysis, the system can be spatially and temporally divided into 'cells' or 'micro-phases' of small (infinitesimal) size, in which classical thermodynamical equilibrium conditions for matter are fulfilled to good approximation. These conditions are unfulfilled, for example, in very rarefied gases, in which molecular collisions are infrequent; and in the boundary layers of a star, where radiation is passing energy to space; and for interacting fermions at very low temperature, where dissipative processes become ineffective. When these 'cells' are defined, one admits that matter and energy may pass freely between contiguous 'cells', slowly enough to leave the 'cells' in their respective individual local thermodynamic equilibria with respect to intensive variables.

''Local thermodynamic equilibrium of matter''<ref name="Gyarmati 1970"/><ref name="G&P 1971"/><ref name="Balescu 1975"/><ref name="Mihalas Mihalas 1984"/><ref name="Schloegl 1989"/> (see also Keizer (1987)<ref name="Keizer 1987"/> means that conceptually, for study and analysis, the system can be spatially and temporally divided into 'cells' or 'micro-phases' of small (infinitesimal) size, in which classical thermodynamical equilibrium conditions for matter are fulfilled to good approximation. These conditions are unfulfilled, for example, in very rarefied gases, in which molecular collisions are infrequent; and in the boundary layers of a star, where radiation is passing energy to space; and for interacting fermions at very low temperature, where dissipative processes become ineffective. When these 'cells' are defined, one admits that matter and energy may pass freely between contiguous 'cells', slowly enough to leave the 'cells' in their respective individual local thermodynamic equilibria with respect to intensive variables.

===Milne's definition in terms of radiative equilibrium===

===Milne's definition in terms of radiative equilibrium===

[[Edward Arthur Milne|Edward A. Milne]], thinking about stars, gave a definition of 'local thermodynamic equilibrium' in terms of the [[thermal radiation]] of the [[matter]] in each small local 'cell'.<ref name="Milne 1928">{{cite journal | last1= Milne |first1= E.A. |year=1928 | title= The effect of collisions on monochromatic radiative equilibrium |journal=[[Monthly Notices of the Royal Astronomical Society]] | volume= 88|issue= 6 |pages=493–502|bibcode=1928MNRAS..88..493M | doi = 10.1093/mnras/88.6.493 |doi-access= free }}</ref> He defined 'local thermodynamic equilibrium' in a 'cell' by requiring that it macroscopically absorb and spontaneously emit radiation as if it were in radiative equilibrium in a cavity at the [[temperature]] of the matter of the 'cell'. Then it strictly obeys Kirchhoff's law of equality of radiative emissivity and absorptivity, with a black body source function. The key to local thermodynamic equilibrium here is that the rate of collisions of ponderable matter particles such as molecules should far exceed the rates of creation and annihilation of photons.

[[Edward Arthur Milne|Edward A. Milne]], thinking about stars, gave a definition of 'local thermodynamic equilibrium' in terms of the [[thermal radiation]] of the [[matter]] in each small local 'cell'.<ref name="Milne 1928">{{cite journal | last1= Milne |first1= E.A. |year=1928 | title= The effect of collisions on monochromatic radiative equilibrium |journal=[[Monthly Notices of the Royal Astronomical Society]] | volume= 88|issue= 6 |pages=493–502|bibcode=1928MNRAS..88..493M | doi = 10.1093/mnras/88.6.493 |doi-access= free }}</ref> He defined 'local thermodynamic equilibrium' in a 'cell' by requiring that it macroscopically absorb and spontaneously emit radiation as if it were in radiative equilibrium in a cavity at the [[temperature]] of the matter of the 'cell'. Then it strictly obeys Kirchhoff's law of equality of radiative emissivity and absorptivity, with a black body source function. The key to local thermodynamic equilibrium here is that the rate of collisions of ponderable matter particles such as molecules should far exceed the rates of creation and annihilation of photons.

==Entropy in evolving systems==

==Entropy in evolving systems==

It is pointed out by W.T. Grandy Jr,<ref>{{cite journal | doi = 10.1023/B:FOOP.0000012007.06843.ed | title = Time Evolution in Macroscopic Systems. I. Equations of Motion | year = 2004 | last1 = Grandy | first1 = W.T., Jr. | journal = Foundations of Physics | volume = 34 | issue = 1 | page = 1 |url=http://physics.uwyo.edu/~tgrandy/evolve.html |arxiv = cond-mat/0303290 |bibcode = 2004FoPh...34....1G }}</ref><ref>{{cite journal | url=http://physics.uwyo.edu/~tgrandy/entropy.html | doi=10.1023/B:FOOP.0000012008.36856.c1 | title=Time Evolution in Macroscopic Systems. II. The Entropy | year=2004 | last1=Grandy | first1=W.T., Jr. | journal=Foundations of Physics | volume=34 | issue=1 | page=21 |arxiv = cond-mat/0303291 |bibcode = 2004FoPh...34...21G }}</ref><ref>{{cite journal | url=http://physics.uwyo.edu/~tgrandy/applications.html | doi = 10.1023/B:FOOP.0000022187.45866.81 | title=Time Evolution in Macroscopic Systems. III: Selected Applications | year=2004 | last1=Grandy | first1=W. T., Jr | journal=Foundations of Physics | volume=34 | issue=5 | page=771 |bibcode = 2004FoPh...34..771G }}</ref><ref>Grandy 2004 see also [http://physics.uwyo.edu/~tgrandy/Statistical_Mechanics.html].</ref>, that entropy, though it may be defined for a non-equilibrium system is—when strictly considered—only a macroscopic quantity that refers to the whole system, and is not a dynamical variable and in general does not act as a local potential that describes local physical forces. Under special circumstances, however, one can metaphorically think as if the thermal variables behaved like local physical forces. The approximation that constitutes classical irreversible thermodynamics is built on this metaphoric thinking.

It is pointed out by W.T. Grandy Jr,<ref>{{cite journal | doi = 10.1023/B:FOOP.0000012007.06843.ed | title = Time Evolution in Macroscopic Systems. I. Equations of Motion | year = 2004 | last1 = Grandy | first1 = W.T., Jr. | journal = Foundations of Physics | volume = 34 | issue = 1 | page = 1 |url=http://physics.uwyo.edu/~tgrandy/evolve.html |arxiv = cond-mat/0303290 |bibcode = 2004FoPh...34....1G }}</ref><ref>{{cite journal | url=http://physics.uwyo.edu/~tgrandy/entropy.html | doi=10.1023/B:FOOP.0000012008.36856.c1 | title=Time Evolution in Macroscopic Systems. II. The Entropy | year=2004 | last1=Grandy | first1=W.T., Jr. | journal=Foundations of Physics | volume=34 | issue=1 | page=21 |arxiv = cond-mat/0303291 |bibcode = 2004FoPh...34...21G }}</ref><ref>{{cite journal | url=http://physics.uwyo.edu/~tgrandy/applications.html | doi = 10.1023/B:FOOP.0000022187.45866.81 | title=Time Evolution in Macroscopic Systems. III: Selected Applications | year=2004 | last1=Grandy | first1=W. T., Jr | journal=Foundations of Physics | volume=34 | issue=5 | page=771 |bibcode = 2004FoPh...34..771G }}</ref><ref>Grandy 2004 see also [http://physics.uwyo.edu/~tgrandy/Statistical_Mechanics.html].</ref>, that entropy, though it may be defined for a non-equilibrium system is—when strictly considered—only a macroscopic quantity that refers to the whole system, and is not a dynamical variable and in general does not act as a local potential that describes local physical forces. Under special circumstances, however, one can metaphorically think as if the thermal variables behaved like local physical forces. The approximation that constitutes classical irreversible thermodynamics is built on this metaphoric thinking.

===Entropy in non-equilibrium===

===Entropy in non-equilibrium===

==Flows and forces==

==Flows and forces==

The fundamental relation of classical equilibrium thermodynamics <ref name="W. Greiner et. al. 1997">W. Greiner, L. Neise, and H. Stöcker (1997), ''Thermodynamics and Statistical Mechanics (Classical Theoretical Physics)'' ,Springer-Verlag, New York, '''P85, 91, 101,108,116''', {{ISBN|0-387-94299-8}}.</ref>

The fundamental relation of classical equilibrium thermodynamics <ref name="W. Greiner et. al. 1997">W. Greiner, L. Neise, and H. Stöcker (1997), ''Thermodynamics and Statistical Mechanics (Classical Theoretical Physics)'' ,Springer-Verlag, New York, '''P85, 91, 101,108,116''', {{ISBN|0-387-94299-8}}.</ref>

===Onsager reciprocal relations===

===Onsager reciprocal relations===

{{Main article|Onsager reciprocal relations}}

{{Main article|Onsager reciprocal relations}}

==Speculated extremal principles for non-equilibrium processes==

==Speculated extremal principles for non-equilibrium processes==

{{main article|Extremal principles in non-equilibrium thermodynamics}}

{{main article|Extremal principles in non-equilibrium thermodynamics}}

==Applications==

==Applications==

Non-equilibrium thermodynamics has been successfully applied to describe biological processes such as [[protein folding]]/unfolding and [[membrane transport|transport through membranes]]<ref>{{cite journal |last1=Kimizuka |first1=Hideo |last2=Kaibara |first2=Kozue |title=Nonequilibrium thermodynamics of ion transport through membranes |journal=Journal of Colloid and Interface Science |date=September 1975 |volume=52 |issue=3 |pages=516–525 |doi=10.1016/0021-9797(75)90276-3}}</ref><ref>{{cite journal |last1=Baranowski |first1=B. |title=Non-equilibrium thermodynamics as applied to membrane transport |journal=Journal of Membrane Science |date=April 1991 |volume=57 |issue=2–3 |pages=119–159 |doi=10.1016/S0376-7388(00)80675-4}}</ref>.

Non-equilibrium thermodynamics has been successfully applied to describe biological processes such as [[protein folding]]/unfolding and [[membrane transport|transport through membranes]]<ref>{{cite journal |last1=Kimizuka |first1=Hideo |last2=Kaibara |first2=Kozue |title=Nonequilibrium thermodynamics of ion transport through membranes |journal=Journal of Colloid and Interface Science |date=September 1975 |volume=52 |issue=3 |pages=516–525 |doi=10.1016/0021-9797(75)90276-3}}</ref><ref>{{cite journal |last1=Baranowski |first1=B. |title=Non-equilibrium thermodynamics as applied to membrane transport |journal=Journal of Membrane Science |date=April 1991 |volume=57 |issue=2–3 |pages=119–159 |doi=10.1016/S0376-7388(00)80675-4}}</ref>.

==See also==

==See also==

==References==

==References==

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{{Reflist|colwidth=30em}}

===Sources===

===Sources===

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==Further reading==

==Further reading==

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==External links==

==External links==

*[https://web.archive.org/web/20110406071945/http://www-dcf.ds.mpg.de/build.php/Titel/Research_english.html?sub=1&ver=en Stephan Herminghaus' Dynamics of Complex Fluids Department at the Max Planck Institute for Dynamics and Self Organization]

*[https://web.archive.org/web/20110406071945/http://www-dcf.ds.mpg.de/build.php/Titel/Research_english.html?sub=1&ver=en Stephan Herminghaus' Dynamics of Complex Fluids Department at the Max Planck Institute for Dynamics and Self Organization]