## Abstract

The problem of irreversibility is difficult and part of this difficulty is due to dealing with the statistical mechanics of a large number of particles. The ergodic theory, founded on the link between thermodynamics and its statistical probability, introduced the ergodic theorem that consists of the equality of microcanonical phase average and the time average of the observables. Moreover, a global approach has been introduced, starting from non-equilibrium thermodynamics and obtaining a general principle of investigation, the principle of maximum entropy generation for the open systems. Here, a general approach to the investigation of irreversible systems and the application of entropy generation to the study of the dynamical systems will be developed.

## 1. Introduction

The variational method is very important in mathematical and theoretical physics because it allows us to describe natural systems by physical quantities independently from the frame of reference used (Özisik 1980). Moreover, the Lagrangian formulation can be used in a variety of physical phenomena, and a structural analogy between different physical phenomena has been pointed out (Truesdell 1970). The most important result of the variational principle consists of obtaining both local and global theories (Truesdell 1970; Lucia 1995): global theory allows us to obtain information directly about the mean values of the physical quantities, while the local one yields information about their distribution (Lucia 1995, 1998, 2001, 2007).

The notions of entropy and its production in equilibrium and non-equilibrium processes form the basis of modern thermodynamics and statistical physics (Dewar 2003; Bruers 2006; Martyushev & Seleznev 2006; Maes & Tasaki 2007; Wang 2007). Entropy has been proved to be a quantity that describes the progress of a non-equilibrium dissipative process. A great contribution was made by Clausius who, in 1854–1862, introduced the notion of entropy in physics, and by Prigogine who, in 1947, proved the minimum entropy production principle (Martyushev & Seleznev 2006). Furthermore, the maximum entropy production principle (MEPP) has been introduced and used by several scientists throughout the twentieth century in their studies of the general theoretical issues of thermodynamics and statistical physics. By this principle, a non-equilibrium system develops following the thermodynamic path that maximizes its entropy production under present constraints (Lucia 1995; Martyushev & Seleznev 2006). The second law of thermodynamics states that for an arbitrary adiabatic process the entropy of the final state is larger than or equal to that of the initial state, which, in terms of the entropy production, represents a new additional statement meaning that the entropy production tends to a maximum. Considering the statistical interpretation of the entropy, it not only tends to increase, but will also increase to a maximum value. Consequently, the MEPP may be viewed as the natural generalization of the Clausius–Boltzmann–Gibbs formulation of the second law (Martyushev & Seleznev 2006). ‘The relationship between the minimum entropy production principle and MEPP is not simple; in fact, these variation principles are absolutely different: although the extremum of one and the same function, the entropy production, is sought, these principles include different constraints and different variable parameters. As a consequence, these principles should not be mutually opposed since they are applicable to different stages of the evolution of a non-equilibrium system’ (Martyushev & Seleznev 2006). It has been proved (Lucia 1995, 2007; Dewar 2003; Bruers 2006; Martyushev & Seleznev 2006; Maes & Tasaki 2007; Wang 2007) that the MEPP, rather than the Prigogine principle, can be a universal principle governing the evolution of non-equilibrium dissipative systems (Lucia 1995; Martyushev & Seleznev 2006). In 2007, the principle of *maximum entropy generation* has been proved for open systems (Lucia 1995, 1998, 2001, 2007), but its statistical expression, which would be useful for dynamical systems analysis has not yet been introduced for a general open system. However, the relation between *entropy production* and *entropy generation*, elsewhere called *entropy variation due to irreversibility* or *irreversible entropy S*_{irr}, has been largely discussed in Lucia (1995).

In addition, all of the developments in the statistical analysis of the system have considered only isolated systems (García-Morales & Pellicer 2006). The relation between the probability measurement in the probability space and real statistical facts remains open. The ergodic theory is founded on the link between thermodynamics and its statistical probability. The ergodic theorem consists of the equality of microcanonical phase average and the time average of the observables (van Lith 2001). Following van Lith, ergodicity means ‘metrical transitivity’.

The problem of irreversibility is difficult and part of the difficulty is due to dealing with the statistical mechanics of a large number of particles (Ruelle 1996). Moreover, a global approach was introduced in 1995 (Lucia 1995), providing a general principle of investigation, the maximum of the entropy generation (Lucia 2007).

Theoretical and mathematical physics study idealized systems and one of the open problems is the understanding of how real systems are related to their idealization. The aim of this paper is to obtain a general approach to the analysis of irreversible systems and to develop the application of entropy generation to the study of the dynamical systems, to obtain a link from statistical and global approaches to open (real) systems. To do this, we will introduce in §2 the thermodynamic open system (real system), in §3 the relation between ergodicity and probability in the system considered, in §4 the entropy generation and its statistical description, in §5 the ergodicity in irreversibility and in §6 the use of the entropy generation in the dynamical systems.

## 2. The system and its measure

In this section the system to be considered will be defined. To do so, we must consider the definition of ‘system with perfect accessibility’, which allows us to define both the thermodynamic system and the dynamical system.

**(****Huang 1987****).** A dynamical state of *N* particles can be specified by the 3*N* canonical coordinates and their conjugate momenta . The 6*N*-dimensional space spanned by is called the phase space *Ω*. A point in the phase space represents a state of the entire *N*-particle system.

**(****Lucia 2001****).** A system with perfect accessibility *Ω*_{PA} is a pair (*Ω*, *Π*), with *Π* a set whose elements *π* are called process generators, together with two functions(2.1)(2.2)where is the state transformation induced by *π*, whose domain (*π*) and range (*π*) are a non-empty subset of *Ω*. This assignment of transformation to process generators is required to satisfy the following conditions of accessibility:

, : the set

*Πσ*is called the set of the states accessible from*σ*and, consequently, it is the entire state space, the phase space*Ω*, andif and .

**(****Lucia 2001****).** A process in *Ω*_{PA} is a pair (*π*, *σ*), with *σ* a state and *π* a process generator such that *σ* is in (*π*). The set of all processes of *Ω*_{PA} is denoted by(2.3)If (*π*, *σ*) is a process, then *σ* is called the initial state for (*π*, *σ*) and *σ* is called the final state for (*π*, *σ*).

**(****Lucia 2001****).** A thermodynamic system is a system with perfect accessibility *Ω*_{PA} with two actions and , called work done and heat gained by the system during the process (*π*, *σ*), respectively.

The set of all these stationary states of a system *Ω*_{PA} is called non-equilibrium ensemble.

**(****Lucia 2001****).** A thermodynamic path *γ* is an oriented piecewise continuously differentiable curve in *Ω*_{PA}.

**(****Lucia 2001****).** A cycle is a path whose end points coincide.

**(****Billingsley 1979****).** A family of subsets of a perfect accessibility phase space *Ω*_{PA} is said to be an algebra if the following conditions are satisfied:

*Ω*_{PA}∈,, and

.

Moreover, it follows that:

,

the algebra is closed under countable intersections and subtraction of sets, and

if

*k*≡{∞} then is said a*σ*-algebra.

**(****Billingsley 1979****).** A function is a measure if it is additive. It means that for any countable subfamily , consisting of mutually disjoint sets, it follows:(2.4)Also, it follows:

,

if and , and

if and .

Moreover, if is a *σ*-algebra and *n*≡{∞}, then the measure is said *σ*-additive.

**(****Gallavotti 2003****).** A smooth map of a compact manifold is a map with the property that around each point *σ* a system of coordinates can be established, based on smooth surfaces and , with s denoting stable and u unstable, of complementary positive dimension, which are the following.

*Covariant*. . This means that the tangent planes to the coordinates surface at*σ*are mapped over the corresponding planes at*σ*.*Continuous*. , with , depends continuously on*σ*.*Transitivity*. There is a point in a subsystem of*Ω*_{PA}of zero Liouville probability, called attractor, with a dense orbit.

A large number of systems also satisfy the hyperbolic condition: the length of a tangent vector ** v** is amplified by a factor for

*k*>0 under the action of if with

*C*>0 and

*λ*<1. This means that if an observer moves with the point

*σ*it sees the nearby points moving around it as if it was a hyperbolic fixed point. But, in a general approach, it is not necessary for this condition to be introduced (Gallavotti 2002, 2003, 2004).

There exists a statistic *μ*_{PA} describing the asymptotic behaviour of almost all initial data in perfect accessibility phase space *Ω*_{PA} such that, except for a volume zero set of initial data *σ*, it will be(2.5)for all continuous functions *φ* on *Ω*_{PA} and for every transformation . For hyperbolic systems, the distribution *μ*_{PA} is the Sinai–Ruelle–Bowen distribution, SRB distribution or SRB statistics (Gallavotti 1995).

The notation *μ*_{PA}(d*σ*) expresses the possible fractal nature of the support of the distribution *μ* and implies that the probability of finding the dynamical system in the infinitesimal volume d*σ* around *σ* may not be proportional to d*σ*. Consequently, it may not be written as *μ*_{PA}(*σ*)d*σ*, but it needs to be conventionally expressed as *μ*_{PA}(d*σ*). The fractal nature of the phase space is an issue yet under debate (García-Morales & Pellicer 2006), but there is a lot of evidence on it in the low-dimensional systems (see Hoover 1998; García-Morales & Pellicer 2006). Here we want to consider this possibility also.

The triple is a *measure space*, the *Kolmogorov probability space Γ*.

A dynamical law *τ* is a group of measure-preserving automorphisms of the probability space *Γ*.

**(****Berkovitx et al. 2006**

**).**A dynamical system consists of a dynamical law

*τ*on the probability space

*Γ*.

## 3. Ergodic and probability

The mathematical foundation of ergodic theory is to establish a connection between phase average and time average (van Lith 2001). It follows an association between ergodic theory and objective interpretation of probability because the ergodic theory establishes a connection between probability measures and objective features of the real world (van Lith 2001). A probability distribution is *stationary* if it is constant at all fixed points in *Γ*, and it reflects the fact that the system is in a *steady* state. Modern ergodicity is founded on the concept of measure-preserving dynamical systems *Γ*_{d} and on the Birkhoff's ergodic theorem (van Lith 2001). Here, considering §2, we introduce the stochastic processes. To do so, we state the following.

*If* *is finite, then for any integrable function* *, the time average* *on γ is defined for all orbits γ outside of a set* *of measure* . *Furthermore*, *is integrable with* , *wherever it is defined, and with*(3.1)

*A measure-preserving transformation* *is ergodic if and only if, for every integrable function* *, the time average* *on γ is equal to the space average* *for all points σ outside of some subset* *of measure* .

Consequently, measurements results as the infinite time averages of phase functions because they take a long time compared with the microscopic relaxation time and, for metrically transitive (=ergodic) systems, measurement results are almost always equal to microcanonical averages (van Lith 2001). Moreover, the initial states lead to different paths in phase space, so the averages depend on the initial state.

Following Gallavotti (2003), we extend his results for our system as follows: let be a Markov partition of the phase space *Ω*_{PA}. Let *T* be a time such that the size of the set , with*σ*=1, …, *m*, is so small that the physical observables are found to be constant inside *E*(*σ*). The probability of *E*(*σ*) and the Liouville distribution *μ*_{L} are described in terms of the functions(3.2)where and are the Jacobian of the unstable and stable manifold, respectively, through *σ* and mapping it to the unstable and stable manifold through *σ*. Moreover, let it be:(3.3)selecting a point for each , Liouville distribution *μ*_{L}, on the phase space *Ω*_{PA}, with volume *V*(*Ω*_{PA}), attributes to the non-empty sets *E*(*σ*) the probability(3.4)(3.5)where *V*(*E*) is the Liouville volume of *E* and and are functions bounded as *σ*, *T* vary. The sets *E*(*σ*) represent macroscopic states, small enough to consider the physical observables to be constant inside them, so that it is possible to introduce the following definition of stability.

**(***ϵ***-Steady state).** Let *Γ*_{d} be a dynamical system and be fixed and non-zero. An open system is in *ϵ*-steady state during the time interval if and only if, for all , there exists such that for all *t*∈, it follows:(3.6)

As a consequence of this last definition, the probability may fluctuate within small bounds and, consequently, dynamical evolution towards steady states is allowed.

**(****Primas 1999****).** An algebraic structure of events is a Boolean algebra such that

if ∈ is an event, then is the event that does not take place,

the element is the event that occurs when at least one of the events and occurs,

the element is the event that occurs when the events and occur,

**1**represents the sure event,**0**represents the impossible element, andif and are any elements of the Boolean algebra , the relation or the equivalent , it follows that implies , represented as .

**(****Primas 1999****).** The mathematical probability theory is the study of a pair (, *p*), where the algebra of events is a *σ*-complete Boolean algebra and the map is a *σ*-additive probability measure. This last map is defined as a norm such that at every event, ∈ is associated a probability *p*() for the occurrence of the event itself. Its properties are the following:

with , where

**0**is the zero of ,*p*(**1**)=1, where**1**is the unit of ,if , then , and

and .

**(****Primas 1999****).** Every probability space *Γ* generates probability algebra with the *σ*-complete Boolean algebra , where *Δ* is the *σ*-ideal of Borel sets (a collection of sets that are considered to be ‘small’ or ‘negligible’) of *μ*_{PA}-measure zero and the restriction of *μ*_{PA} to is a strictly positive measure *p*. Conversely, every probability algebra (, *p*) can be realized by some Kolmogorov probability space *Γ*.

**(****Primas 1999****).** A statement is said to be true *almost everywhere* or *for almost all σ* if it is true for all except in a set of measure zero, .

**(****Primas 1999****).** von Neumann proved that the finest empirically accessible events are given by Borel sets of non-vanishing Lebesgue measure, and not by the much larger class of all subsets of *Ω*_{PA}.

**(****Primas 1999****).** The Borel *σ*-algebra of subsets of the set of real numbers is the *σ*-algebra generated by the open subsets of . In a Kolmogorov's set-theoretical formulation, a statistical observable is a *σ*-homomorphism . Every observable *ξ* can be induced by a real-valued Borel function by means of the inverse map(3.7)A real-valued Borel function *π* defined on *Ω*_{PA} is said to be real-valued random variable.

**(****Primas 1999****).** Every statistical observable is induced by a random variable, while an observable, that is a *σ*-homomorphism, defines only an equivalence class of random variables that induce this homomorphism.

**(****Primas 1999****).** For a statistical description, it is not necessary to know the point function , but it is sufficient to know the observable *ξ*. The description of a physical system in terms of an individual function distinguishes between different points and corresponds to an individual description of equivalence classes of random variables, which do not distinguish between different points and corresponds to a statistical ensemble description.

Let it be a real-valued Borel function such that is integrable over *Ω*_{PA} with respect to *μ*_{PA}, the expectation value *π*_{ev} of (*σ*) with respect to *μ*_{PA} is(3.8)

**(****Primas 1999****).** Every Borel-measurable complex-valued function of a random variable on *Γ* is also a complex-valued random variable on *Γ*.

**(****Primas 1999****).** A real-valued random variable on *Γ* induces a probability measurement on the state space . Considering the probability as a property of the generating conditions of a sequence, randomness can be related to predictability and retrodictability (Primas 1999). A family is called a stochastic process, which can be represented by a family of equivalent classes of random variables *ξ*(*t*) on *Γ*. The point function *γ*(*σ*(*t*)) is called the trajectory of the stochastic process *ξ*(*t*). The description of physical systems in terms of a trajectory of a stochastic process corresponds to a point dynamics, while its description in terms of equivalent classes of trajectories and an associated probability measure corresponds to an ensemble dynamics (Primas 1999).

A stochastic process is said weakly stationary if

,

, and

.

**(****Primas 1999****).** As a consequence of the Wiener–Khintchine theorem (Primas 1999), there exists a complex-valued function , continuous at the origin, which is the covariance function of a complex-valued, second-order, weakly stationary and continuous stochastic process if and only if it can be represented as(3.9)where, as a consequence of the Bochner–Cramér representation theorem (Primas 1999), is a real, never decreasing and bounded function called spectral distribution function of the stochastic process. Moreover, the Lebesgue's decomposition theorem states that the spectral distribution function can be decomposed uniquely as(3.10)with *c*^{d}≥0, *c*^{s}≥0, *c*^{ac}≥0, *c*^{d}+*c*^{s}+*c*^{ac}=1, step function, continuous and singular real function, absolutely continuous real function.

There exists a close relationship between regular and stochastic processes and the irreversibility. A system is called irreversible if the lost energy is strictly positive. Following König and Tobergte (Primas 1999), a linear input–output system behaves irreversible if and only if the associated distribution function fulfils the Wiener–Krein criterion for the spectral density of a linear regular stochastic process: a weakly stationary stochastic process *ξ* with mean value *ξ*_{ev}=0 and spectral distribution is linearly regular if and only if its spectral distribution function is absolutely continuous and if(3.11)

Following Wiener and Akutowics (Primas 1999), we obtain the following.

Every stationary process with absolutely continuous spectral function (=regular process) is ergodic.

In 1932, Koopman and von Neumann stated the mixing theorem about which it is useful to consider the statement of Halmos (1958): ‘a geometric property of *T* (mixing) is equivalent to a spectral property of *U* (no non-trivial proper values). In somewhat informal but quite suggestive language, the mixing theorem says that a necessary and sufficient condition that each pair of sets be eventually stochastically independent is that the spectrum be essentially continuous. There is also a beautiful result about the very opposite situation in which *U* has pure point spectrum (i.e. *L*_{2} has an orthonormal basis consisting of proper vectors of *U*). For ergodic transformations of this kind, it turns out that the spectrum, which is always a subset of the set of complex numbers of modulus 1, is in fact a subgroup of the multiplicative group of such numbers, and every such subgroup is the spectrum of some such transformation. If both *S* and *T* satisfy the conditions (ergodic, pure point spectrum), then a necessary and sufficient condition for the measure-theoretic isomorphism of *S* and *T* is the unitary equivalence of the corresponding unitary operators. In other words, for a special but large class of transformations, the analytic (operator) methods give complete information about the geometric (transformation) questions’. Consequently, if is the past, then the future is completely determined if the Szegö condition for perfect linear predictability is fulfilled (Primas 1999),(3.12)Moreover, every function with an absolutely continuous spectral distribution fulfilling the Paley–Wiener criterion (Primas 1999)(3.13)is called a chaotic function in the sense of Wiener. The link between Wiener's and Kolmogorov's statistical approaches is Birkhoff's ergodic theorem (corollary 3.2); in fact, it implies the existence of a *μ*_{PA}-almost every trajectory of an ergodic stochastic process on the space *Γ*.

## 4. Entropy generation

Theoretical and mathematical physics study idealized systems and one of the open problems is the understanding of how real systems are related to their idealization. In the analysis of irreversibility, the concept of thermostats has been introduced: they are systems of particles moving outside the system and interacting with the particles of the system through interactions across the walls of the system itself. In the statistical and dynamical approaches to thermodynamics, the fundamental quantity considered by Gallavotti (2003) is the entropy production *σ*_{entr}, which is defined as follows:(4.1)with always and only for the equilibrium state and , with *f*_{int}(*σ*) the internal conservation force, *f*_{nc}(*σ*) the external active non-conservative force and *f*_{term}(*σ*) the thermostatic expression. Moreover, the thermostat temperature *T*_{term} is defined as(4.2)with mass flow.

**(****Lucia 1995****).** Following Truesdell (1970), for each continuum thermodynamic system (isolated, closed or open), in which it is possible to identify a thermodynamic subsystem with elementary mass d*m* and elementary volume d*V*=d*m*/*ρ*, with *ρ* the mass density (Truesdell 1970; Lucia 1995, 1998, 2001, 2007), the thermodynamic description can be developed by referring to the generalized coordinates , with , *α*_{i} the extensive thermodynamic quantities and their values at the stable states.

In thermodynamic engineering, the energy lost for irreversible processes is evaluated by the first and second law of thermodynamics for the open systems (Lucia 1998). So the following definition can be introduced.

The entropy generation is defined as(4.3)where *W*_{lost} is the work lost for irreversibility; *T*_{ref} the temperature of the lower reservoir; *Q*_{r} the heat source; *T*_{r} its temperature; *T*_{a} the ambient temperature; *H* the enthalpy; *S* the entropy; *E*_{k} the kinetic energy; *E*_{g} the gravitational one; and *W* the work. The quantities *W*_{lost}, *Q*_{r}, *H*, *S*, *E*_{k}, *E*_{g} and *W* are expressed per unit mass flow.

**(****Lucia** **1995****,** **1998****,** **2001****,** **2007****).** *The thermodynamic Lagrangian can be obtained as*(4.4)

For every subsystem, a thermodynamic Lagrangian per unit time *t*, temperature *T* and volume *V*, is defined as (Lucia 1995, 1998, 2001, 2007)(4.5)where (Lucia 1995)(4.6)is the entropy per unit time and volume, and *ψ* is the nonlinear dissipative potential density defined as (Lucia 1995)(4.7)with *L*_{ij} Onsager coefficients defined as (Gallavotti 2006)(4.8)where *σ*_{α}, *α*=*i*, *j* are the α-state and are the *r* external forces. Consequently, becomes(4.9)which is also the Legendre transformation to the relation (4.7) (Lucia 1995). Moreover, the thermodynamic Lagrangian is defined as(4.10)and considering that (Lucia 1995)(4.11)where *ρ*_{S} is the entropy per unit time and mass and *ρ*_{π} is the power per unit mass and temperature. Now, following Lavenda (Lucia 2001), , so that (Lucia 2001, 2007)(4.12)and(4.13)so it follows:(4.14)but remembering thatwe can obtain(4.16)Considering the Gouy–Stodola theorem (Lucia 1995), which states that(4.17)the relation (4.10) becomes(4.18)with *T*_{ref} the temperature of the lower reservoir and *S*_{irr} the entropy generation (Lucia 2001, 2007). ▪

As a consequence of the definition of *T*_{ref} and *T*_{term}, we assume that(4.19)

Considering the following relations (4.2), (4.4), (4.18) and (4.19), it follows that:(4.20)

The statistical expression, for the irreversible-entropy variation, results(4.21)

**(****Lucia 2007****).** * The principle of maximum entropy generation. The condition of stability for* the

*open system' stationary states that its entropy generation*

*reaches its maximum*(4.22)

The thermodynamic action is defined as (Truesdell 1970)(4.23)From the principle of the least action,(4.24)Remembering the relation (4.4), it follows that:(4.25)which becomes(4.26)and if *T*_{ref} is constant,(4.27) ▪

## 5. Irreversible ergodicity

As a consequence of the theorem (4.7) and the relation (4.21), we can obtain the following.

*In non-equilibrium transformation, the volume of the phase space* *contracts indefinitely*.

As a consequence of the relations (4.21) and (4.22) it follows that:(5.1)which becomes(5.2)that is(5.3)Following the results obtained in 1995 (Lucia 1995), starting from the Onsager principle, it follows that:(5.4)so that(5.5)Then, as a consequence of the discussion about the *μ*_{PA}-distribution, the volume of the phase space *Ω*_{PA} contracts indefinitely during a dissipative (i.e. real) transformation. ▪

The phase space cells, which represent the stationary states, form a subset of all the cells on which the evolution acts as a one-cycle permutation. Following Gallavotti (1995, 2000), this is a kind of ergodicity. Now, we can define it ‘ergodicity for irreversibility’.

## 6. Dynamical systems and entropy generation

The evolution of a dynamical system can be obtained by means of the relations (Ruelle 1996)(6.1)

Now, we consider the principle of maximum entropy generation. It represents a general principle of investigation for the stability of the open systems; in fact, it states that: *in a general thermodynamic transformation, the condition of the stability for the open system steady states consists of the maximum for the entropy generation* (see equation (4.22); Lucia 2007). This statement represents an important result in irreversible processes thermodynamics, because it is a global theoretical principle for the analysis of the stability of open system states: during any transformation, open systems follow the path such that the entropy generation becomes maximum (Lucia 2007).

Now, to use the principle of maximum entropy generation in the analysis of dynamical systems, we must introduce the result obtained in §5 in the relation (6.1)(6.2)

One of the problems in the relation (6.1) is to find the right distribution. In the last form (6.2), here obtained, we have overcome this difficulty by introducing the global thermodynamic effect of the action of ** E**, expressed by the second and third equation of the relation (6.2).

## 7. Conclusions

The aim of this paper was to obtain a general approach to the analysis of irreversible systems and to develop the application of the entropy generation to the study of the dynamical systems, to obtain a link from statistical and global approaches to open (real) systems. To do this, we have introduced in §2 the thermodynamic open system (real system), in §3 the relation between ergodicity and probability, in §4 the entropy generation and its statistical description, in §5 the ergodicity in irreversibility and in §6 the use of the entropy generation in the dynamical systems.

In phenomena out of equilibrium, irreversibility manifests itself because the fluctuations of the physical quantities, which bring the system apparently out of stationarity, occur symmetrically about their average values (Gallavotti 2006). The *ϵ*-steady-state definition allows us to obtain that for certain fluctuations the probability of occurrence follows a universal law and the frequency of occurrence is controlled by a quantity that has been related to the entropy generation. Moreover, this last quantity has a purely mechanical interpretation that is related to the ergodic hypothesis, which proposed that an isolated system evolves in time visiting all possible microscopic states. Moreover, considering that the open system is the one with perfect accessibility represented as a probability space in which is defined a PA-measure and a Borel function process, the ergodic hypothesis itself is a consequence of the ϵ-steady-state definition owing to the hypothesis 3.18.

The principle of maximum entropy generation represents the macroscopic effect of these theories that, conversely, are its statistical interpretation.

The results obtained are useful to cope with the analysis of the differential equation system (6.1) in which the difficulty consists of finding the right PA distribution. Using the statistical definition of entropy generation, the relation (6.1) becomes (6.2) in which global quantities are introduced. In this way, it is possible to study the relation (6.1) with a global approach using only global thermodynamic quantities, which can be easily obtained for any system.

As Primas (1999) has pointed out: ‘The importance of the ergodic theorem lies in the fact that in most applications we see only a single individual trajectory that is a particular realization of the thermostatic process. The Kolmogorov's theory of stochastic processes refers to the equivalence classes of functions and the Birkhoff's ergodic theorem provides a crucial link between the ensemble description and the individual description of chaotic phenomena.’ The modern concept of subjective probabilities implies a behaviour based on the Boolean logic (Primas 1999). The result obtained also represents the relation between the probability measurement in the probability space and real statistical facts. Following Primas (1999), the set-theoretical representation of the Boolean algebra in terms of a Kolmogorov probability space is useful because it allows to relate a dynamic in terms of a probabilistic density and all known context-independent physical laws are formulated in terms of pure states (Primas 1999).

Finally, entropy generation has been introduced in the analysis of dynamical systems. In fact, one of the problems in the approach of dynamical system is to find the right distribution. By entropy generation, we have overcome this difficulty introducing the global thermodynamic effect of the action of the forces, expressed by the second and third equations of the relation (6.2), finding a global expression that links the microscopical values to the macroscopical thermodynamic quantities.

## Footnotes

- Received November 5, 2007.
- Accepted January 2, 2008.

- © 2008 The Royal Society