Research inIrreversible processInNonequilibrium stateThe macroscopic theory of thermodynamic phenomena of physical systems.Thermodynamics of irreversible processes is a macro theory, and its explanation of non-equilibrium phenomenon is limited after all.In particular, thermodynamic theory cannot explain the formation mechanism of various complex structures and the systematicFluctuationCharacteristics, which requires a more in-depth theory - nonequilibrium statistical physics (seeStatistical Physics）To complete.
The theory of equilibrium thermodynamics based on equilibrium state and reversible process has been quite perfect and widely used in the macroscopic description of various physical and chemical processes.However, in nature, in physics, chemistry, meteorology, astrophysicslife sciencesAmong many problems involved in the field of environmental ecology, non-equilibrium thermodynamic systems and irreversible processes exist in large quantities.For example, nucleic acids in living cells are constantly exchanging materials with their environment, and the steady flow of energy from the sun makes the earth's atmosphere unable to reach thermodynamic balance.Therefore, it has become an urgent need to extend thermodynamic methods to irreversible processes, and gradually formed a new research field - irreversible process thermodynamics. The corresponding micro theory is non-equilibrium statistical physics.
Local balance
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Non equilibrium physical systems are often very complex, some close to equilibrium, some far from equilibrium.Thermodynamics of irreversible processes usually only discusses non-equilibrium systems satisfying local equilibrium conditions, that is, each local volume element is in equilibrium, and state parameters andThermodynamic functionDescription.But at the same time, these thermodynamic functions are also functions of time and space, so we try to establish their equations of motion. These equations stipulate the nature of local equilibrium but overall non-equilibrium of the system. Because the equations are often nonlinear and difficult to solve, some approximate calculation methods have been developed in recent years, and some valuable results have been obtained.
Introduction to Theory
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For systems satisfying local equilibrium conditions, it can be considered that various thermodynamic functions of equilibrium state are still applicable to local volume elements of non-equilibrium systems, and the relationship between thermodynamic functions also remains valid.Thus, the local entropy should be
Where{ρj}={(ρ1(r,t),ρ2(r,t),ρ3(r,t)…，ρn(r,t）}Represent the composition of various substances in the systemtSpace density at time.If it is an isothermal isobaric system, the relationship between the local entropy and the total entropy of the non-equilibrium system is
。On the other hand, the system is generally unbalanced{ρj}Over timetThe law of change is determined by the law of conservation
Wherefi({ρj}) represents the components in the systemρjRate of change, which is generallyρ1,ρ2,…，ρOf nNonlinear function，DiEu 2 ρ i describes the diffusion process caused by uneven density, and Di is the diffusion coefficient.Equation (3) is often calledReaction diffusion equation。The above all reflect the nature of local balance and overall imbalance of the system. Therefore, under the condition of local balance, these formulasNon-equilibrium thermodynamicsThe characteristics of the system are all specified.The solution of the above equation becomes difficult because of the nonlinear function fi ({ρ}) in equation (3).In addition to using approximate methods to calculate, the stability theory of Riyapunov differential equations is a useful tool.According to Riyapunov's theory, for equation (3), if a function V=V ({ρ j}) can be found, and there is the property of V ≥ 0, dV/dt ≤ 0 near the spatial density {ρ j0} of a stationary state [let V ({ρ j0})=0], then the stationary state is stable;On the contrary, if V ≥ 0 and dV/dt ≥ 0, the steady state is unstable.The function V is called Riapunov function.The basic thermodynamic picture of irreversible processes near and far from the equilibrium region can be obtained concisely by using the stability theory.The near equilibrium region refers to the region near the equilibrium state.It can be considered that the force here is relatively weak, so the relationship between "force" (such as temperature gradient, concentration gradient, etc.) and "flow" (such as heat flow, diffusion flow, etc.) caused by it can be approximately described by linear relationship
WhereJI is some kind of "flow", XjIs the various "forces" that cause this flow, the coefficientLijIt has the following symmetry
This relationship is calledOnsager reciprocal relationship。Therefore, the near equilibrium region is also called the linear non-equilibrium region, or the linear region for short.In the near equilibrium or linear region, due to the nature of "force" and "flow", various specific effects can be caused, but the non-equilibrium system eventually tends to be stable, which is its basic characteristic.This characteristic was expressed by I. Prigokin in the form of minimum entropy production principle.According to Formula (1) and Formula (3), the conservation equation of local entropy in non-equilibrium system is
WherejS is the entropy flow density, which describes the transfer of entropy between the local volume element and the outside world;σ is local entropy production, which describes the increase rate of entropy caused by irreversible processes in local volume elements.It can be proved that under the condition of local equilibrium, the expression of σ is
The total entropy of the system is zero.
fromThe second law of thermodynamicsIt can be seen that σ ≥ 0,P≥0。Using the linear relationship between "flow" and "force" (4) and the Onsager reciprocity relationship (5), it can also be proved that
The above formula indicates that,PIt is a decreasing function, and its value tends to be stable only when it reaches a steady state.In other words, the stationary state is the state in which entropy production takes the minimum value, and this conclusion is the principle of minimum entropy production.From here, we can see that entropy production plays the role of thermodynamic potential in equilibrium theory in the linear non-equilibrium region.At the same time, entropy generation can be used as the Lyapunov function to judge the stability of the system in the linear non-equilibrium region.Equation (9) shows that the system here is always stable, any accidental deviation from the steady state will disappear with time, and the system will return to the original steady state.Therefore, it is impossible to make a sudden change in the linear region - to make the system transition to a new stationary state and present an ordered structure.
The relationship between "flow" and "force" of the non-equilibrium system is usually nonlinear when it is far away from the equilibrium region, so this region is also called the nonlinear region.The problems discussed in irreversible process thermodynamics far away from the equilibrium zone are the possibility of new structure formation, the transformation of disorder and order, etc. These problems are of great significance both in theory and practice.
In the nonlinear area, the change of the system is much more complicated than in the linear area, but there are also certain laws.According to the general discussion of irreversible process thermodynamics, the relationship between
here
Represents the part of total entropy generation that changes with time due to mechanical changes.It can be seen that the total decreases with time.Equation (10) can be written more specifically, that is
This gives
Specific contents included.Its δJi、δXjThey respectively represent the change of "flow" and "force" from the steady state,
It reflects the asymmetric part of the relationship between "flow" and "force" that does not meet equation (4).Equation (10) gives a general criterion for the time varying system in the nonlinear region, which is also applicable to the case of the linear region.In the linear area, from the symmetry of the Onsager relationship and the relationship between "flow" and "force", it can be seen that
, so
Because there is δ x in the linear areaP≥ 0, which satisfies the Lyapunov stability condition, so the system is stable.It is consistent with the previous formula (9).For the general case of nonlinear region
It may be positive, negative or zero.Therefore, the system in the corresponding nonlinear zone is not always stable, and it is possible to achieve a sudden change from stable to unstable.This is the condition for the appearance of ordered structures and other complex shapes.Among the phenomena that these systems are far from equilibrium, people are most interested in self-organization and chaos.To analyze these specific structures and graphs, we need to combine the specific system discussed through equation (3);Solve under certain boundary conditions, and select some typical models to solve equation (3).[1]
application
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In physics and chemical systems, heat conduction, diffusion, conductivity, chemical reaction, etc. are some basic non-equilibrium phenomena. Applying the principle of irreversible process thermodynamics to discuss these phenomena, we can get meaningful concrete results.In some nonequilibrium systems, there are often cross phenomena of many irreversible processes.For example, in the mixture system, when the concentration and temperature are non-uniform, there are heat conduction, diffusion and their cross effects.For these cross effects, thermodynamics of irreversible processes has been well applied in the linear region.In the nonlinear area, this kind of application is more extensive. In addition to physical and chemical systems, it can also be applied to life systems and ecological balance and other issues.At present, we mainly discuss several typical systems such as fluid, laser, electronic circuit, chemical reaction and ecology.The main issues discussed are the formation of self-organized orderly structure, the classification of graphsNonequilibrium phase transitionAnd the self similar structure in chaos.The study of these phenomena enriches the thermodynamics of irreversible processes.
The concepts of fluctuation theory and stochastic process play an important role in the discussion of such basic problems.Onsager reciprocity relation can be obtained from the statistical properties of multi particle system.Stochastic process theory can be used to discuss the relationship between the spontaneous fluctuation of the system and the macroscopic response of the system under the external strengthening force.For the phenomenon far from equilibrium, we can use the stochastic process theory to discuss the phase transition of the non-equilibrium system caused by the amplification of fluctuations, which leads to the generation of new structures.These can deepen the understanding of non-equilibrium phenomena (seedissipative structure ）。
The near equilibrium region refers to the region near the equilibrium state.Here, the force, such as temperature gradient and concentration gradient, is weak, and there is approximately a linear relationship between them and the resulting flow, such as heat flow and diffusion flow. Therefore, it is also called linear non-equilibrium zone, whose basic feature is that the non-equilibrium system will eventually tend to balance.In the linear region, some cross effects (such as the cross effects of heat conduction and diffusion) in non-equilibrium systems have been well explained by irreversible process thermodynamics.
Far from equilibriumNon-equilibrium thermodynamicsIn the system, the relationship between force and flow is nonlinear, also called nonlinear region.The system in the nonlinear zone does not always tend to be stable, but instead may realize a sudden change from stable to unstable, which is other complex graphs of ordered structure andNonequilibrium phase transitionAmong them, people are most interested in self-organization, chaos and other phenomena. The study of these problems is of great significance in theory and practice.[2]
