The thermodynamic hamiltonian for open systems Umberto Lucia 1 1 I.T.I.S. “A. Volta”, Spalto Marengo 42, 15121 Alessandria, Italy 0 2 n Abstract a J The variational method is very important in mathematical and 6 theoretical physics because it allows us to describe the natural sys- tems by physical quantities independently from the frame of reference ] h used. A global and statistical approach have been introduced start- p ing from non-equilibrium thermodynamics, obtaining the principle of - h maximumentropy generation fortheopensystems. Thisprincipleisa at consequence of the lagrangian approach to the open systems. Here it m will be developed a general approach to obtain the thermodynamic [ hamiltonian for the dynamical study of the open systems. It fol- lows that the irreversibility seems to bethe fundamental phenomenon 1 v which drives the evolution of the states of the open systems. 2 Keywords: dynamical systems, entropy, non-equilibrium thermo- 1 3 dynamics, rational thermodynamics, irreversibility 1 . 1 1 Introduction 0 1 1 : The variational method is very important in mathematical and theoretical v physics because it allows usto describe thenatural systems by physical quan- i X titiesindependently fromtheframeofreferenceused(Ozisik1980). Moreover r a 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 in 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 about their distribution (Lucia 1995 - Lucia 2007). The notions of entropy and its production in equilibrium and non- equilibrium processes form the basis of modern thermodynamics and sta- tistical physics (Bruers 2006, Dewar 2003, Maes & Tasaki 2007, Martyushev 1 & Seleznev 2006, Wang 2007). Entropy has been proved to be a quantity describing the progress of non-equilibrium dissipative process. Great con- tribution has been done in this 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). A Lagrangian approach to this subject allowed to obtained the mathematical consequences on the behaviour of the entropy generation, elsewhere called entropy variation due to irreversibility or irreversible entropy S (Lucia irr 2008). 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 rational thermodynamic approach to the analysis of irreversible systems and to de- velop the application of the entropy generation to the study of these systems by the dynamical approach. To do this we will introduce in Section 2 the thermodynamic open system (real system), in Section 3 the relation between the thermodynamics Lagrangian and the entropy generation, in Section 4 the thermodynamic hamiltonian and the hamiltonian approach to thermo- dynamics. 2 The system considered In this section it will be defined the system considered. 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. Let us consider an open continuum or discrete N particles system. Every i−th element ofthissystemislocatedbyapositionvectorx ∈ R3, ithasavelocity i x˙ ∈ R3, a mass m ∈ R and a momentum p = m x˙ , with i ∈ [1,N] and i i i i p ∈ R3 (Lucia 1995, Lucia 1998, Lucia 2001). The masses m must satisfy i the condition: N m = m (1) i i=1 X where m is the total mass which must be a conserved quantity so that it follows: ρ˙ +ρ∇·x˙ = 0 (2) B where ρ = dm/dV is the total mass density, with V total volume of the system and x˙ ∈ R3, defined as x˙ = N p /m, velocity of the centre of B B i=1 i mass. The mass density must satisfy the following conservation law (Lucia P 2 1995, Lucia 1998, Lucia 2001): ρ˙ +ρ ∇·x˙ = ρΞ (3) i i i N where ρ is the density of the i−th elementary volume V , with V = V, i i i=1 i and Ξ is the source, generated by matter transfer, chemical reactions and P thermodynamic transformations. This open system can be mathematical defined as follows (Lucia 2008). Definition 1 (Huang 1987) - A dynamical state of N particles can be spec- ified by the 3N canonical coordinates {q ∈ R3,i ∈ [1,N]} and their conju- i gate momenta {p ∈ R3,i ∈ [1,N]}. The 6N−dimensional space spanned by i {(p ,q ),i ∈ [1,N]} is called the phase space Ω. A point σ = (p ,q ) in i i i i i i∈[1,N] the phase space Ω := σ ∈ R6N : σ = (p ,q ),i ∈ [1,N] represents a state i i i i of the entire N−particle system. (cid:8) (cid:9) Definition 2 (Lucia 2001) - A system with perfect accessibility Ω is a pair PA (Ω,Π), with Π a set whose elements π are called process generators, together with two functions: π 7→ S (4) ′ ′′ ′′ ′ π ,π 7→ π π (5) (cid:16) (cid:17) where S is the state transformation induced by π, whose domain D(π) and range R(π) are non-empty subset of Ω. This assignment of transformation to process generators is required to satisfy the following conditions of acces- sibility: 1. Πσ := {Sσ : π ∈ Π,σ ∈ D(π)} = Ω , ∀σ ∈ Ω: the set Πσ is called the set of the states accessible from σ and, consequently, it is the entire state space, the phase spase Ω; 2. if D(π′′)∩R(π′) 6= 0 ⇒ D(π′′π′) = Sπ−′1(D(π′′)) and Sπ′′π′σ = Sπ′′Sπ′σ ∀σ ∈ D(π′′π′) Definition 3 (Lucia 2001) - A process in Ω is a pair (π,σ), with σ a state PA and π a process generator such that σ is in D(π). The set of all processes of Ω is denoted by: PA Π⋄Ω = {(π,σ) : π ∈ Π,σ ∈ D(π)} (6) If (π,σ) is a process, then σ is called the initial state for (π,σ) and Sσ is called the final state for (π,σ). 3 Definition 4 (Lucia 2001) - A thermodynamic system is a system with per- fect accessibility Ω with two actions W (π,σ) ∈ R and H(π,σ) ∈ R, called PA work done and heat gained by the system during the process (π,σ), respec- tively. Comment 1 (Lucia2008) - The set of all these stationary states of a system Ω is called non-equilibrium ensemble. PA Definition 5 (Lucia 2001) - A thermodynamic path γ is an oriented piece- wise continuously differentiable curve in Ω . PA Definition 6 (Lucia 2001) - A cycle C is a path whose endpoints coincide. Definition 7 (Billingsley 1979) - A family F of subset of a perfect accessi- bility phase-space Ω is said to be an algebra if the following conditions are PA satisfied: 1. Ω ∈ F PA 2. Ω0 ∈ F ⇒ Ω0c ∈ F PA PA k 3. {Ω } ⊂ F ⇒ Ω ∈ F PAi i∈[1,k] i=1 PAi S Comment 2 (Lucia 2008) - Moreover, it follows that: 1. ∅ ∈ F 2. the algebra F is closed under countable intersections and subtraction of sets 3. if k ≡ {∞} then F is said a σ−algebra Definition 8 (Billingsley 1979) - A function µ : F → [0,∞) is a measure if it is additive. It means that for any countable subfamily {Ω ,i ∈ [1,n]} ⊆ F, i consisting of mutually disjoint sets, it follows: n n µ Ω = µ(Ω ) (7) i i ! i=1 i=1 [ X It follows, too: 1. µ(∅) = 0 2. if Ω ,Ω ∈ F and Ω ⊂ Ω ⇒ µ(Ω ) ≤ µ(Ω ) α β α β α β 4 n 3. if Ω ⊂ Ω ⊂ ··· ⊂ Ω and {Ω ,i ∈ [1,n]} ∈ F ⇒ µ( Ω ) = 1 2 n i i=1 i sup µ(Ω ). i i S Moreover, if F is a σ−algebra and n ≡ {∞} then the measure is said σ−additive. Definition 9 (Gallavotti 2003) - A smooth map S of a compact manifold M is a map with the property that around each point σ it can be established a system of coordinates based on smooth surfaces Ws and Wu, with s=stable σ σ and u=unstable, of complementary positive dimension which is: 1. covariant: ∂SWi = Wi ,i = u,s. This means that the tangent planes σ Sσ ∂SWi,i = u,s to the coordinates surface at σ are mapped over the σ corresponding planes at Sσ; 2. continuous: ∂SWi, with i = u,s, depends continuously on σ; σ 3. transitivity: there is a point in a subsistem of Ω of zero Liouville PA probability, called attractor, with a dense orbit. Comment 3 (Lucia 2008) A great number of systems satisfies also the hy- perbolic condition: the lenght of a tangent vector v is amplified by a factor Cλk for k > 0 under the action of S−k if σ ∈ Ws with C > 0 and λ < 1. This k 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 this condition is not necessary to be introduced. Hypothesis 1 (Lucia 2008) - There exists a statistics µ describing the PA asymptotic behaviour of almost all initial data in perfect accessibility phase space Ω such that, except for a volume zero set of initial data σ, it will be: PA T−1 1 lim ϕ Sjσ = µ (dσ)ϕ(σ) (8) PA T−→∞ T j=1 ZΩ X (cid:0) (cid:1) for all continuous functions ϕ on Ω and for every transformation σ 7→ PA S (σ). For hyperbolic systems the distribution µ is the Sinai-Ruelle-Bowen t PA distribution, SRB-distribution or SRB-statistics (Gallavotti 1995). Comment 4 (Lucia 2008) - The notation µ (dσ) expresses the possible PA fractal nature of the support of the distribution µ, and implies that the prob- ability of finding the dynamical system in the infinitesimal volume dσ around σ may not be proportional to dσ. Consequently, it may not be written as 5 µ (σ)dσ, but it needs to be conventionally expressed as µ (dσ). The frac- PA PA tal nature of the phase space is an issue yet under debate (Garc`Ia-Morales & Pellicer 2006), but there are a lot of evidence on it in the low dimensional systems (see ref. Garc`Ia-Morales & Pellicer 2006 and Hoover 1998). Here we want to consider also this possibility. Definition 10 (Lucia 2008) - The triple (Ω ,F,µ ) is a measure space, PA PA the Kolmogorov probability space Γ. Definition 11 (Lucia 2008) - A dynamical law τ is a group of meausure- preserving automorphisms S : Ω → Ω of the probability space Γ. PA PA Definition 12 (Berkovitz et al. 2006) - A dynamical system Γ = d (Ω ,F,µ ,τ) consists of a dynamical law τ on the probability space Γ. PA PA 3 Thermodynamic Lagrangian and 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 it has been introduced the concept of thermostats, which are systems of particles moving outside the system and interacting with the particles of the system through interactions acrossthewallsofthesystemitself. Inthestatisticalanddynamicalapproach tothermodynamics, thefundamental quantity considered by Gallavottiisthe entropy production σ , defined as follows (Gallavotti 2003): entr σ = k σentr = −k µ (dσ)∇·E(σ) (9) entr B + B PA ZΩ with always σentr ≥ 0 and σentr = 0 only for the equilibrium state and + + E(σ) := f (σ)+f (σ)+f (σ), with f (σ) internal consevation force, int nc term int f (σ) external active non conservative force and f (σ) thermostatic ex- nc term pression. Comment 5 (Lucia 1995) - Following Truesdell (Truesdell 1970), for each continuum thermodynamic system (isolated or closed or open), in which it is possible to identify a thermodynamics subsystem with elementary mass dm and elementaryvolume dV = dm/ρ, with ρ mass density (Truesdell 1970,Lu- cia 1995, Lucia 1997, Lucia 2001, Lucia 2007) the thermodynamic description ˙ can be developed by referring to the generalized coordinates ξ ,ξ ,t , i i i∈[1,N] n o 6 (0) (0) with ξ = α −α , α the extensive thermodynamic quantities and α their i i i i i 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: Definition 13 (Lucia 2008) - The entropy generation ∆S is defined as: irr W Q T ∆H ∆E +∆E −W lost r a k g = ∆S := 1− + −∆S + (10) irr T T T T T ref a r a a (cid:18) (cid:19) where W is the work lost for irreversibility, T the temperature of the lost ref lower reservoir, Q is the heat source, T its temperature, T is the ambient r r a temperature, H is the entalpy, S is the entropy, E is the kinetic energy, E k g is the gravitational one and W is the work. Comment 6 (Lucia 2008) - It has been proved that ∆S = m˙ σ , with m˙ irr entr mass flow. Theorem 1 (Lucia 1995, Lucia 1997, Lucia 2001, Lucia 2007, Lucia 2008) - The termodynamic Lagrangian can be obtained as: L = −T ∆S (11) ref irr Proof. For every subsystem a termodynamic Lagrangian per unit time t, temperature T and volume V, ρ is defined as (Lucia 1995, Lucia 1997, L Lucia 2001, Lucia 2007): d3L dS˙ ρ = = −ψ (12) L dVdTdt dV where (Lucia 1995): dS˙ 1 = L ξ ξ + L ξ ξ ξ (13) ij i j ijk i j k dV 2 ij ijk X X is the entropy per unit time and volume, and ψ is the non-linear dissipative potential density, defined as (Lucia 1995): 1 1 ψ = L ξ ξ + L ξ ξ ξ (14) ij i j ijk i j k 2 6 ij ijk X X 7 with L Onsager coefficients, defined as (Gallavotti 2006): ij ∞ 1 L = µ [σ (S σ)σ (σ)] dt (15) ij 2 PA i t j E=0 −∞ Z where σ , α = i,j, are the α-state and E = {E } are the r external α k k∈[1,r] forces. Consequently, ρ becomes: L 1 1 ρ = L ξ ξ + L ξ ξ ξ (16) L ij i j ijk i j k 2 3 ij ijk X X which is the Legendre transformation to the relation (14), too (Lucia 1995). Moreover, the thermodynamic Lagrangian L is defined as: L := dt dT dVρ (17) L Zt ZT ZV and considering that (Lucia 1995): ρ = ρ −ρ −ψ (18) L S π where ρ is the entropy per unit time and mass and ρ is the power per unit S π mass and temperature. Now, following Lavenda (Lucia 2001), ρ −ρ = 2ψ, S π so that (Lucia 2001, Lucia 2007): ρ = ψ (19) L and dt dT dVρ = dt dT dVψ (20) L Zt ZT ZV Zt ZT ZV so it follows: L = dt dT dVψ (21) Zt ZT ZV but, remembering that: W := dt dT dVψ (22) lost Zt ZT ZV we can obtain: L = W (23) lost Considering the Gouy-Stodola theorem (Lucia 1995), which states that: W = −T ∆S (24) lost ref irr 8 the (11) becomes: L = dt dT dVψ = −T ∆S (25) ref irr Zt ZT ZV with T the temperature of the lower reservoir and S the entropy gener- ref irr ation (Lucia 2001, Lucia 2007). Definition 14 (Lucia 2008) The statistical expression, for the irreversible- entropy variation, results: k B ∆S = − µ (dσ)∇·E(σ) (26) irr PA m˙ ZΩ Theorem 2 (Lucia 2007) - The principle of maximum entropy gen- eration (Lucia 2007): The condition of stability for the open system’ sta- tionary states is that its entropy generation ∆S reaches its maximum: irr δ(∆S ) ≥ 0 (27) irr Proof. The thermodynamic action is defined as (Truesdell 1970): A := dtL (28) Zt From the principle of the least action: δA ≤ 0 (29) remebering the relation (11) it follows that: −δ dtT ∆S ≤ 0 (30) ref irr (cid:18)Z (cid:19) which becomes: δ(T ∆S ) ≥ 0 (31) ref irr and if T is constant ref δ(∆S ) ≥ 0 (32) irr 9 4 Thermodynamic Hamiltonian and entropy generation Theorem 3 - The thermodynamic hamiltonian: The thermodynamic Hamiltonian can be obtain as T ∆S . ref irr Proof. The thermodynamic Hamiltonian density ρ can be defined following H the definition of the thermodynamic lagrangian density ρ : L ρ = ζ ξ −ρ (33) H i i L i X with ∂ρ L ζ = (34) i ˙ ∂ξ i coniugate generalised momentum to thegeneralised coordinatesξ . Thether- i modynamic lagrangian density ρ is defined by the relation (12) and it is a L function only of the generalised coordinates, so that it follows that: ∂ρ L = 0 (35) ˙ ∂ξ i Consequently, the thermodynamic Hamiltonian density results: ρ = −ρ (36) H L and the thermodynamic Hamiltonian can be obtained as: H = dt dT dVρ = − dt dT dVρ = H L (37) Zt ZT ZV Zt ZT ZV = −L = T ∆S ref irr Comment 7 From the definition of action it follows that the thermodynamic action can be written as: A = dtL = − dtH (38) Zt Zt Comment 8 The thermodynamic hamiltonian for open systems is related only to the entropy generation. Consequently, this quantity seems to be the basis of the analysis of these systems. Comment 9 Moreover, the irreversibility seems to be the fundamental phe- nomenon which drives the evolution of the states of the open systems. 10