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Non-equilibrium thermodynamics
 
 
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Non-equilibrium thermodynamics is a branch of thermodynamics
Thermodynamics

In physics, thermodynamics is the study of the conversion of heat energy into different forms of energy ; different energy conversions into heat energy; and its relation to macroscopic variables such as temperature, pressure, and volume....
 concerned with studying time
Time

Time is a component of the measurement used to sequence events, to compare the durations of events and the intervals between them, and to quantify the motions of objects....
-dependent thermodynamic system
Thermodynamic system

In thermodynamics, a thermodynamic system, originally called a working substance, is defined as that part of the universe that is under consideration....
s, irreversible transformations and open systems. In contrast to most physical theories that rely on thermodynamic equilibrium, most systems found in nature are not in equilibrium. Real systems are not isolated from their environment and are therefore continuously sharing energy with other systems. This sharing of energy includes being driven by external energy sources as well as dissipating energy.

Non-equilibrium systems cannot be studied with all the tools of equilibrium thermodynamics such as entropy production, the equipartition of energy, the definition of temperature or predicting the flow of heat.






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Non-equilibrium thermodynamics is a branch of thermodynamics
Thermodynamics

In physics, thermodynamics is the study of the conversion of heat energy into different forms of energy ; different energy conversions into heat energy; and its relation to macroscopic variables such as temperature, pressure, and volume....
 concerned with studying time
Time

Time is a component of the measurement used to sequence events, to compare the durations of events and the intervals between them, and to quantify the motions of objects....
-dependent thermodynamic system
Thermodynamic system

In thermodynamics, a thermodynamic system, originally called a working substance, is defined as that part of the universe that is under consideration....
s, irreversible transformations and open systems. In contrast to most physical theories that rely on thermodynamic equilibrium, most systems found in nature are not in equilibrium. Real systems are not isolated from their environment and are therefore continuously sharing energy with other systems. This sharing of energy includes being driven by external energy sources as well as dissipating energy.

Non-equilibrium systems cannot be studied with all the tools of equilibrium thermodynamics such as entropy production, the equipartition of energy, the definition of temperature or predicting the flow of heat. Systems driven out of equilibrium do often exhibit patterned behavior such as fluctuations or phase transitions. Non-equilibrium thermodynamics, as contrasted with equilibrium thermodynamics
Equilibrium thermodynamics

Equilibrium Thermodynamics is the systematic study of transformations of matter and energy in systems as they approach equilibrium. The word equilibrium implies a state of balance....
, is most successful in the study of steady state
Steady state

A system in a steady state has numerous properties that are unchanging in time. The concept of steady state has relevance in many fields, in particular thermodynamics....
s, where there are nonzero forces, flows and entropy
Entropy

In many branches of science, entropy is a measure of the disorder of a system. The concept of entropy is particularly notable as it is applied across physics, information theory and mathematics....
 production, but no time variation
Time-variant system

A time-variant system is a system, that is not time-invariant....
. One of the early non-equilibrium systems to be discovered was the Belousov-Zhabotinsky
Belousov-Zhabotinsky reaction

A Belousov?Zhabotinsky reaction, or BZ reaction, is one of a class of reactions that serve as a classical example of non-equilibrium thermodynamics, resulting in the establishment of a nonlinear chemical clock....
 chemical oscillator.

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 thermodynamic process. Here a strong temperature difference is maintained between two molecular degrees of freedom (with molecular laser, vibrational and rotational molecular motion). Damping of acoustical perturbations or shock waves are non-stationary non-equilibrium processes. Driven complex fluids
Complex fluids

Complex fluids are binary mixtures that have a coexistence between two Phase s: solid-liquid , solid-gas , liquid-gas and liquid-liquid . They exhibit unusual mechanical responses to applied Stress or Strain due to the geometrical constraints that the phase coexistence imposes....
, turbulent systems and glasses are other examples of non-equilibrium systems.

The mechanics of macroscopic systems depends on a number of extensive quantities. It should be stressed that all systems are permanently interacting with their surroundings, thereby causing unavoidable fluctuations of extensive quantities. Equilibrium conditions of thermodynamic systems are related to the maximum property of the entropy. If the only extensive quantity that is allowed to fluctuate is the internal energy, all the other ones being kept strictly constant, the temperature of the system is measurable and meaningful. The system's properties are then most conveniently described using the thermodynamic potential Helmholtz free energy
Helmholtz free energy

In thermodynamics, the Helmholtz free energy is a thermodynamic potential which measures the ?useful? work obtainable from a closed system thermodynamic thermodynamic system at a constant temperature and volume....
 (A = U - TS), a Legendre transformation
Legendre transformation

In mathematics, it is often desirable to express a functional relationship as a different function, whose argument is the derivative of f , rather than x ....
 of the energy. If, next to fluctuations of the energy, the macroscopic dimensions (volume) of the system are left fluctuating, we use the Gibbs free energy
Gibbs free energy

In thermodynamics, the Gibbs free energy is a thermodynamic potential that measures the "useful" or process-initiating Work obtainable from an isothermal, Isobaric process thermodynamic system....
 (G = U + PV - TS), where the system's properties are determined both by the temperature and by the pressure. Non-equilibrium systems are much more complex and they may undergo fluctuations of more extensive quantities. The boundaries conditions impose to them particular intensive variables, like temperature gradients or distorted collective motions (shear motions, vortices, etc), often called thermodynamic forces. If free energies are very useful in equilibrium thermodynamics, it must be stressed that there is no general law defining stationary non-equilibrium properties of the energy as is the second law of thermodynamics for the entropy
Entropy

In many branches of science, entropy is a measure of the disorder of a system. The concept of entropy is particularly notable as it is applied across physics, information theory and mathematics....
 in equilibrium thermodynamics. That is why in such cases a more generalized Legendre transformation should be considered. This is the extended Massieu potential. By definition, the entropy
Entropy

In many branches of science, entropy is a measure of the disorder of a system. The concept of entropy is particularly notable as it is applied across physics, information theory and mathematics....
 (S) is a function of the collection of extensive quantities . Each extensive quantity has a conjugate intensive variable (a restricted definition of intensive variable is used here by comparison to the definition in given in this link) so that:

We then define the extended Massieu function as follows:

where is Boltzmann's constant, whence

The independent variables are the intensities.

Intensities are global values, valid for the system as a whole. When boundaries impose to the system different local conditions, (e.g. temperature differences), there are intensive variables representing the average value and others representing gradients or higher moments. The latter are the thermodynamic forces driving fluxes of extensive properties through the system.

It may be shown that the Legendre transformation changes the maximum condition of the entropy (valid at equilibrium) in a minimum condition of the extended Massieu function for stationary states, no matter whether at equilibrium or not.

Flows and forces

The fundamental relation of thermodynamics



expresses the change in entropy
Entropy

In many branches of science, entropy is a measure of the disorder of a system. The concept of entropy is particularly notable as it is applied across physics, information theory and mathematics....
  of a system as a function of the intensive quantities temperature
Temperature

In physics, temperature is a physical property of a Physical system that underlies the common notions of hot and cold; something that feels hotter generally has the greater temperature....
 , pressure
Pressure

Pressure is the force per unit area applied to an object in a direction surface normal to the surface. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure....
  and chemical potential
Chemical potential

In thermodynamics, physics and chemistry, chemical potential, symbolized by ?, is a term introduced by the American engineer, chemist and mathematical physicist Willard Gibbs, which he defined as follows:...
  and of the differentials of the extensive quantities energy
Energy

In physics, energy is a scalar physical quantity that describes the amount of Work_ that can be performed by a force. Energy is an attribute of objects and systems that is subject to a conservation law....
 , volume
Volume

The volume of any solid, liquid, plasma, vacuum or theoretical object is how much three-dimensional space it occupies, often quantified numerically....
  and particle number
Particle number

The particle number, N, is the number of constituent particles in a Thermodynamics. The particle number is a fundamental parameter in thermodynamics and it is Conjugate variables to the chemical potential....
 .

Using , and as our basis of the extensive quantities, we then see that the corresponding thermodynamic forces are the gradients of , and respectively. In stationary conditions, the extensive quantities are by definition constant. We may therefore state that the system's entropy is equally constant. However, according to Ilya Prigogine
Ilya Prigogine

Ilya, Viscount Prigogine was a Russian-born naturalization Belgium chemist and Nobel Prize noted for his work on dissipative system, complex systems, and irreversibility....
 and others , when an open system is allowed to reach a stationary state, it organizes itself so as to minimize total entropy production. In fact, entropy production does not occur within the system but it is located at the boundaries where the system exchanges its extensive properties with the neighbourhood.

The intensive variables representing gradients of properties are thermodynamic forces driving fluxes of the associated extensive quantities. Establishing the relation between flows of extensive quantities and the intensive variables representing gradients is a problem in statistical mechanics. Flows may be coupled. This is expressed by Onsager's equations
Onsager reciprocal relations

In thermodynamics, the Onsager reciprocal relations express the equality of certain relations between fluxs and forces in thermodynamic systems out of equilibrium , but where a notion of local thermodynamic equilibrium exists....
. In the regime where both the flows are small and the thermodynamic forces vary slowly, there will be a linear relation between them, parametrized by a matrix
Matrix (mathematics)

In mathematics, a matrix is a rectangular array of numbers, as shown at the right. In addition to a number of elementary, entrywise operations such as matrix addition a key notion is matrix multiplication....
 of coefficients conventionally denoted :

The Onsager relations

In the regime where both the flows are small and the thermodynamic forces vary slowly, there will be a linear relation between them, parametrized by a matrix
Matrix (mathematics)

In mathematics, a matrix is a rectangular array of numbers, as shown at the right. In addition to a number of elementary, entrywise operations such as matrix addition a key notion is matrix multiplication....
 of coefficients conventionally denoted :

The second law of thermodynamics
Second law of thermodynamics

The second law of thermodynamics is an expression of the universal law of increasing entropy, stating that the entropy of an isolated system which is not in Thermodynamic equilibrium will tend to increase over time, approaching a maximum value at equilibrium....
 requires that the matrix be positive definite
Positive definite

In mathematics, positive definite may refer to:* positive-definite matrix* positive-definite function** positive definite function on a group...
. Statistical mechanics
Statistical mechanics

Statistical mechanics is the application of probability theory, which includes Mathematics tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force....
 considerations involving microscopic reversibility of dynamics imply that the matrix is symmetric
Symmetric matrix

In linear algebra, a symmetric matrix is a square matrix, A, that is equal to its transposeThe entries of a symmetric matrix are symmetric with respect to the main diagonal ....
. This fact is called the Onsager reciprocal relations.

Microscopic reversibility of the dynamics implies that the matrix is symmetrical (Onsagers reciprocal relations).

Stationary states and the principle of minimal entropy production


See also

  • Maxwell's daemon
  • Information entropy
    Information entropy

    In information theory, entropy is a measure of the uncertainty associated with a random variable. The term by itself in this context usually refers to the Shannon entropy, which quantifies, in the sense of an expected value, the self-information contained in a message, usually in units such as bits....
  • Self-organization
    Self-organization

    Self-organization is a process of attraction and VSEPR theory in which the internal organization of a system, normally an open system , increases in complexity without being guided or managed by an outside source....
  • Self-organizing criticality
  • Autocatalytic reactions and order creation


External links

  • - 1992- book by Xavier de Hemptinne.
  • - PhysicsToday.org
  • - 2005 book by Dorion Sagan and Eric D. Schneider, on nonequilibrium thermodynamics and evolutionary theory
    Evolution

    In biology, evolution is change in the heritability trait of a population of organisms from one generation to the next. These changes are caused by a combination of three main processes: variation, reproduction, and selection....
    .