THERMODYNAMIC MODELING, ENERGY EQUIPARTITION, AND NONCONSERVATION OF ENTROPY FOR DISCRETE-TIME DYNAMICAL SYSTEMS WASSIM M. HADDAD, QING HUI, SERGEY G. NERSESOV, AND VIJAYSEKHAR CHELLABOINA Received 19 November 2004 We develop thermodynamic models for discrete-time large-scale dynamical systems. Specifically, using compartmental dynamical system theory, we develop energy flow mod- els possessing energy conservation, energy equipartition, temperature equipartition, and entropy nonconservation principles for discrete-time, large-scale dynamical systems. Fur- thermore, we introduce a new and dual notion to entropy; namely, ectropy, as a measure of the tendency of a dynamical system to do useful work and grow more organized, a nd show that conservation of energy in an isolated thermodynamic system necessarily leads to nonconservation of ect ropy and entropy. In addition, using the system ectropy as a Lya- punov function candidate, we show that our discrete-time, large-scale thermodynamic energy flow model has convergent trajectories to Lyapunov stable equilibria determined by the system initial subsystem energies. 1. Introduction Thermodynamic principles have been repeatedly used in continuous-time dynamical sys- tem theor y as well as in information theory for developing models that capture the ex- change of nonnegative quantities (e.g., mass and energy) between coupled subsystems [5, 6, 8, 11, 20, 23, 24]. In particular, conservation laws (e.g., mass and energy) are used to capture the exchange of material between coupled macroscopic subsystems known as compartments. Each compartment is assumed to be kinetically homogeneous; that is, any material entering the compartment is instantaneously mixed with the material in the compartment. These models are known as compartmental models and are widespread in engineering systems as well as in biological and ecological sciences [1, 7, 9, 16, 17, 22]. Even though the compartmental models developed in the literature are based on the first law of thermodynamics involving conservation of energy principles, they do not tell us whether any particular process can actually occur; that is, they do not address the second law of thermodynamics involving entropy notions in the energy flow between subsys- tems. The goal of the present paper is directed towards developing nonlinear discrete-time compartmental models that are consistent with thermodynamic principles. Specifically, Copyright © 2005 Hindawi Publishing Corporation Advances in Difference Equations 2005:3 (2005) 275–318 DOI: 10.1155/ADE.2005.275 276 Thermodynamic modeling for discrete-time systems since ther modynamic models are concerned with energy flow among subsystems, we develop a nonlinear compartmental dynamical system model that is characterized by en- ergy conservation laws capturing the exchange of energy between coupled macroscopic subsystems. Furthermore, using graph-theoretic notions, we state three thermodynamic axioms consistent with the zeroth and second laws of thermodynamics that ensure that our large-scale dynamical system model gives rise to a thermodynamically consistent en- ergy flow model. Specifically, using a large-scale dynamical systems theory perspective, we show that our compartmental dynamical system model l eads to a precise formula- tion of the equivalence between work energy and heat in a large-scale dynamical sys- tem. Next, we give a deterministic definition of entropy for a large-scale dynamical sys- tem that is consistent with the classical thermodynamic definition of entropy and show that it satisfies a Clausius-type inequality leading to the law of entropy nonconservation. Furthermore, we introduce a new and dual notion to entropy; namely, ectropy,asamea- sure of the tendency of a large-scale dynamical system to do useful work and grow more organized, and show that conservation of energy in an isolated thermodynamically con- sistent system necessarily leads to nonconservation of ectropy and entropy. Then, using the system ectropy as a Lyapunov function candidate, we show that our thermodynami- cally consistent large-scale nonlinear dynamical system model possesses a continuum of equilibria and is semistable ; that is, it has convergent subsystem energies to Lyapunov sta- ble energy equilibria determined by the large-scale s ystem initial subsystem energies. In addition, we show that the steady-state distribution of the large-scale system energies is uniform leading to system energy equipartitioning corresponding to a minimum ectropy and a maximum entropy equilibrium state. In the case where the subsystem energies are proportional to subsystem temperatures, we show that our dynamical system model leads to temperature equipartition, wherein all the system energy is transferred into heat at a uniform temperature. Furthermore, we show that our system-theoretic definition of entropy and the newly proposed notion of ectropy are consistent with Boltzmann’s kinetic theory of gases involving an n-body theory of ideal gases divided by diather mal walls. The contents of the paper are as follows. In Section 2, we establish notation, defi- nitions, and rev iew some basic results on nonnegative and compartmental dynamical systems. In Section 3, we use a large-scale dynamical systems perspective to develop a nonlinear compartmental dynamical system model characterized by energy conservation laws that is consistent with basic thermodynamic principles. Then we tur n our attention to stability and convergence. In particular, using the total subsystem energies as a candi- date system energy storage function, we show that our thermodynamic system is lossless and hence can deliver to its surroundings all of its stored subsystem energies and can store all of the work done to all of its subsystems. Next, using the system ectropy as a Lyapunov function candidate, we show that the proposed thermodynamic model is semistable with a uniform energy distribution corresponding to a minimum ectropy and a maximum en- tropy. In Section 4, we generalize the results of Section 3 to the case where the subsystem energies in large-scale dynamical system model are proportional to subsystem tempera- tures and arrive at temperature equipartition for the proposed thermodynamic model. Wassim M. Haddad et al. 277 Furthermore, we provide an interpretation of the steady-state expressions for entropy and ectropy that is consistent with kinetic theory. In Section 5, we specialize the results of Section 3 to ther modynamic models with linear energy exchange. Finally, we draw con- clusions in Section 6. 2. Mathematical preliminaries In this section, we introduce notation, several definitions, and some key results needed for developing the main results of this paper. Let R denote the set of real numbers, let Z + denote the set of nonnegative integers, let R n denote the set of n × 1columnvectors,let R m×n denote the set of m × n real matrices, let (·) T denote transpose, and let I n or I denote the n×n identity matrix. For v ∈ R q ,wewritev ≥≥ 0(resp.,v  0) to indicate that every component of v is nonnegative (resp., positive). In this case, we say that v is nonnegative or positive, respectively. Let R q + and R q + denote the nonnegative and positive orthants of R q ; that is, if v ∈ R q ,thenv ∈ R q + and v ∈ R q + are equivalent, respectively, to v ≥≥ 0and v  0. Finally, we write ·for the Euclidean vector norm, ᏾(M)andᏺ(M)forthe range space and the null space of a matrix M, respectively, spec(M) for the spectrum of the square mat rix M,rank(M) for the rank of the matrix M,ind(M) for the index of M; that is, min{k ∈ Z + :rank(M k ) = rank(M k+1 )}, M # for the group generalized inverse of M, where ind(M) ≤ 1, ∆E(x(k)) for E(x(k +1))− E(x(k)), Ꮾ ε (α), α ∈ R n , ε>0, for the open ball centered at α with radius ε,andM ≥ 0(resp.,M>0) to denote the fact that the Hermitian matrix M is nonnegative (resp., positive) definite. The following definition introduces the notion of Z-, M-, nonnegative, and compart- mental matrices. Definit ion 2.1 [2, 5, 12]. Let W ∈ R q×q . W is a Z-matrix if W (i, j) ≤ 0, i, j = 1, ,q, i = j. W is an M-matrix (resp., a nonsingular M-matrix)ifW is a Z-matrix and all the principal minors of W are nonnegative (resp., positive). W is nonnegative (resp., positive)ifW (i,j) ≥ 0(resp.,W (i, j) > 0), i, j = 1, ,q.Finally,W is compartmental if W is nonnegative and  q i=1 W (i, j) ≤ 1, j = 1, ,q. In this paper, it is important to distinguish between a square nonnegative (resp., posi- tive) matrix and a nonnegative-definite (resp., positive-definite) matrix. The following definition introduces the notion of nonnegative functions [12]. Definit ion 2.2. Let w = [w 1 , ,w q ] T : ᐂ → R q ,whereᐂ is an open subset of R q that con- tains R q + .Thenw is nonnegative if w i (z) ≥ 0foralli = 1, , q and z ∈ R q + . Note that if w(z) = Wz,whereW ∈ R q×q ,thenw(·) is nonnegative if and only if W is a nonnegative matrix. Proposition 2.3 [12]. Suppose that R q + ⊂ ᐂ. Then R q + is an invariant set with respect to z(k +1)= w  z(k)  , z(0) = z 0 , k ∈ Z + , (2.1) where z 0 ∈ R q + ,ifandonlyifw : ᐂ → R q is nonnegative. 278 Thermodynamic modeling for discrete-time systems The following definition introduces several types of stability for the discrete-time nonnegative dynamical system (2.1). Definit ion 2.4. The equilibrium solution z(k) ≡ z e of (2.1)isLyapunov stable if, for every ε>0, there exists δ = δ(ε) > 0suchthatifz 0 ∈ Ꮾ δ (z e ) ∩ R q + ,thenz(k) ∈ Ꮾ ε (z e ) ∩ R q + , k ∈ Z + . The equilibrium solution z(k) ≡ z e of (2.1)issemistable if it is Lyapunov stable and there exists δ>0suchthatifz 0 ∈ Ꮾ δ (z e ) ∩ R q + , then lim k→∞ z(k) exists and corresponds to a Lyapunov stable equilibrium point. The equilibrium solution z(k) ≡ z e of (2.1)isasymptotically stable if it is Lyapunov stable and there exists δ>0suchthatif z 0 ∈ Ꮾ δ (z e ) ∩ R q + , then lim k→∞ z(k) = z e . Finally, the equilibrium solution z(k) ≡ z e of (2.1)isglobally asymptotically stable if the previous statement holds for all z 0 ∈ R q + . Finally, recall that a matrix W ∈ R q×q is semistable if and only if lim k→∞ W k exists [12], while W is asymptotically stable if and only if lim k→∞ W k = 0. 3. Thermodynamic modeling for discrete-time systems 3.1. Conservation of energy and the first law of thermodynamics. The fundamental and unifying concept in the analysis of complex (large-scale) dynamical systems is the concept of energy. The energy of a state of a dynamical system is the measure of its abil- ity to produce changes (motion) in its own system state as well as changes in the system states of its surroundings. These changes occur as a direct consequence of the energy flow between different subsystems within the dynamical system. Since heat (energy) is a funda- mental concept of thermodynamics involving the capacity of hot bodies (more energetic subsystems) to produce work, thermodynamics is a theory of large-scale dynamical sys- tems [13]. As in thermodynamic systems, dynamical systems can exhibit energy (due to friction) that becomes unavailable to do useful work. This is in turn contributes to an increase in system entropy; a measure of the tendency of a system to lose the ability to do useful work. To develop discrete-time compartmental models that are consistent with thermody- namic principles, consider the discrete-time large-scale dynamical system Ᏻ shown in Figure 3.1 involving q interconnected subsystems. Let E i : Z + → R + denote the energy (and hence a nonnegative quantity) of the ith subsystem, let S i : Z + → R denote the ex- ternal energy supplied to (or extrac ted from) the ith subsystem, let σ ij : R q + → R + , i = j, i, j = 1, ,q, denote the exchange of energy from the jth subsystem to the ith subsystem, and let σ ii : R q + → R + , i = 1, , q, denote the energy loss from the ith subsystem. An energ y balance equation for the ith subsystem yields ∆E i (k) = q  j=1, j=i  σ ij  E(k)  − σ ji  E(k)  − σ ii  E(k)  + S i (k), k ≥ k 0 , (3.1) or, equivalently, in vector form, E(k +1)= w  E(k)  − d  E(k)  + S(k), k ≥ k 0 , (3.2) Wassim M. Haddad et al. 279 S 1 S i S j S q Ᏻ 1 Ᏻ i Ᏻ j Ᏻ q . . . . . . σ 11 (E) σ ii (E) σ jj (E) σ qq (E) σ ij (E) σ ji (E) Figure 3.1. Large-scale dynamical system Ᏻ. where E(k) = [E 1 (k), ,E q (k)] T , S(k) = [S 1 (k), ,S q (k)] T , d(E(k)) = [σ 11 (E(k)), , σ qq (E(k))] T , k ≥ k 0 ,andw = [w 1 , ,w q ] T : R q + → R q is such that w i (E) = E i + q  j=1, j=i  σ ij (E) − σ ji (E)  , E ∈ R q + . (3.3) Equation (3.1) yields a conservation of energy equation and implies that the change of energy stored in the ith subsystem is equal to the external energy supplied to (or ext racted from) the ith subsystem plus the energ y gained by the ith subsystem from all other sub- systems due to subsystem coupling minus the energy dissipated from the ith subsystem. Note that (3.2) or, equivalently, (3.1) is a statement reminiscent of the first law of thermo- dynamics for each of the subsystems, with E i (·), S i (·), σ ij (·), i = j,andσ ii (·), i = 1, ,q, playing the role of the ith subsystem internal energy, energy supplied to (or extracted from) the ith subsystem, the energy exchange between subsystems due to coupling, and the energy dissipated to the environment, respectively. To further elucidate that (3.2) is essentially the statement of the principle of the con- servation of energy, let the total energy in the discrete-time large-scale dynamical system Ᏻ be g iven by U  e T E, E ∈ R q + ,wheree T  [1, ,1], and let the energy received by the discrete-time large-scale dynamical system Ᏻ (in forms other than work) over the discrete-time interval {k 1 , ,k 2 } be given by Q   k 2 k=k 1 e T [S(k) − d(E(k))], where E(k), k ≥ k 0 , is the solution to (3.2). Then, premultiplying (3.2)bye T and using the fact that e T w(E) ≡ e T E, it follows that ∆U = Q, (3.4) 280 Thermodynamic modeling for discrete-time systems where ∆U  U(k 2 ) − U(k 1 ) denotes the variation in the total energy of the discrete-time large-scale dynamical system Ᏻ over the discrete-time interval {k 1 , ,k 2 }. This is a state- ment of the first law of thermodynamics for t he discrete-time large-scale dynamical sys- tem Ᏻ and gives a precise formulation of the equivalence between variation in system internal energy and heat. It is impor t ant to note that our discrete-time large-scale dynamical system model does not consider work done by the system on the environment nor work done by the envi- ronment on the system. Hence, Q can be interpreted physically as the amount of energy that is received by the system in forms other than work. The extension of addressing work performed by and on the system can be easily handled by including an additional state equation, coupled to the energy balance equation (3.2), involving volume states for each subsystem [13]. Since this slight extension does not alter any of the results of the paper, it is not considered here for simplicity of exposition. For our large-scale dynamical system model Ᏻ, we assume that σ ij (E) = 0, E ∈ R q + , whenever E j = 0, i, j = 1, ,q. This constraint implies that if the energy of the jth sub- system of Ᏻ is zero, then this subsystem cannot supply any energy to its surroundings nor dissipate energ y to the environment. Furthermore, for the remainder of this paper, we as- sume that E i ≥ σ ii (E) − S i −  q j=1, j=i [σ ij (E) − σ ji (E)] =−∆E i , E ∈ R q + , S ∈ R q , i = 1, ,q. This constraint implies that the energy that can be dissipated, extr a cted, or exchanged by the ith subsystem cannot exceed the current energy in the subsystem. Note that this as- sumption implies that E(k) ≥≥ 0forallk ≥ k 0 . Next, premultiplying (3.2)bye T and using the fact that e T w(E) ≡ e T E, it follows that e T E  k 1  = e T E  k 0  + k 1 −1  k=k 0 e T S(k) − k 1 −1  k=k 0 e T d  E(k)  , k 1 ≥ k 0 . (3.5) Now, for the discrete-time large-scale dynamical system Ᏻ, define the input u(k)  S(k) and the output y(k)  d(E(k)). Hence, it follows from (3.5) that the discrete-time large- scale dynamical system Ᏻ is lossless [23] with respect to the energy supply rate r(u, y) = e T u − e T y and with the energy storage function U(E)  e T E, E ∈ R q + . This implies that (see [23] for details) 0 ≤ U a  E 0  = U  E 0  = U r  E 0  < ∞, E 0 ∈ R q + , (3.6) where U a  E 0   − inf u(·),K≥k 0 K−1  k=k 0  e T u(k) − e T y(k)  , U r  E 0   inf u(·),K≥−k 0 +1 k 0 −1  k=−K  e T u(k) − e T y(k)  , (3.7) and E 0 = E(k 0 ) ∈ R q + .SinceU a (E 0 ) is the maximum amount of stored energy which can be extracted from the discrete-time large-scale dynamical system Ᏻ at any discrete-time instant K,andU r (E 0 )istheminimumamountofenergywhichcanbedeliveredto Wassim M. Haddad et al. 281 the discrete-time large-scale dynamical system Ᏻ to transfer it from a state of minimum potential E(−K) = 0toagivenstateE(k 0 ) = E 0 ,itfollowsfrom(3.6) that the discrete- time large-scale dynamical system Ᏻ can deliver to its surroundings all of its stored sub- system energies and can store all of the work done to all of its subsystems. In the case where S(k) ≡ 0, it follows from (3.5) and the fact that σ ii (E) ≥ 0, E ∈ R q + , i = 1, ,q,that thezerosolutionE(k) ≡ 0 of the discrete-time large-scale dynamical system Ᏻ with the energy balance equation (3.2) is Lyapunov stable with Lyapunov function U(E)corre- sponding to the total energy in the system. The next result shows that the large-scale dynamical system Ᏻ is locally controllable. Proposition 3.1. Consider the discrete-time large-scale dynamical system Ᏻ with energy balance equation (3.2). Then for every equilibrium state E e ∈ R q + and every ε>0 and T ∈ Z + , there exist S e ∈ R q , α>0,and  T ∈{0, ,T} such that for every  E ∈ R q + with   E − E e ≤αT,thereexistsS : {0, ,  T}→R q such that S(k) − S e ≤ε, k ∈{0, ,  T},and E(k) = E e +((  E − E e )/  T)k, k ∈{0, ,  T}. Proof. Note that w ith S e = d(E e ) − w(E e )+E e , the state E e ∈ R q + is an equilibrium state of (3.2). Let θ>0andT ∈ Z + ,anddefine M(θ,T)  sup E∈Ꮾ 1 (0),k∈{0, ,T}   w  E e + kθE  − w  E e  − d  E e + kθE  + d  E e  − kθE   . (3.8) Note that for every T ∈ Z + ,lim θ→0 + M(θ,T) = 0. Next, let ε>0andT ∈ Z + be given, and let α>0besuchthatM(α,T)+α ≤ ε. (The existence of such an α is guaranteed since M(α,T) → 0asα → 0 + .) Now, let  E ∈ R q + be such that   E − E e ≤αT.With  T    E − E e /α≤T,wherex denotes the smallest integer greater than or equal to x,and S(k) =−w  E(k)  + d  E(k)  + E(k)+  E − E e     E − E e   /α  , k ∈{0, ,  T}, (3.9) it follows that E(k) = E e +   E − E e      E − E e   /α  k, k ∈{0, ,  T}, (3.10) is a solution to (3.2). The result is now immediate by noting that E(  T) =  E and   S(k) − S e   ≤     w  E e +   E − E e      E − E e   /α  k  − w  E e  − d  E e +   E − E e      E − E e   /α  k  + d  E e  −   E − E e      E − E e   /α  k     + α ≤ M(α,T)+α ≤ ε, k ∈{0, ,  T}. (3.11)  It follows from Proposition 3.1 that the discrete-time large-scale dynamical system Ᏻ with the energy balance equation (3.2)isreachable from and cont rollable to the origin in 282 Thermodynamic modeling for discrete-time systems R q + . Recall that the discrete-time large-scale dynamical system Ᏻ with the energy balance equation (3.2) is reachable from the origin in R q + if, for all E 0 = E(k 0 ) ∈ R q + , there exist a finite time k i ≤ k 0 and an input S(k)definedon{k i , ,k 0 } such that the state E(k), k ≥ k i , canbedrivenfromE(k i ) = 0toE(k 0 ) = E 0 .Alternatively,Ᏻ is controllable to the origin in R q + if, for all E 0 = E(k 0 ) ∈ R q + , there exist a finite time k f ≥ k 0 and an input S(k)definedon {k 0 , ,k f } such that the state E(k), k ≥ k 0 , can be driven from E(k 0 ) = E 0 to E(k f ) = 0. We let ᐁ r denote the set of all admissible bounded energy inputs to the discrete-time large-scale dynamical system Ᏻ such that for any K ≥−k 0 , the system energy state can be driven from E(−K) = 0toE(k 0 ) = E 0 ∈ R q + by S(·) ∈ ᐁ r ,andweletᐁ c denote the set of all admissible bounded energy inputs to the discrete-time large-scale dynamical system Ᏻ such that for any K ≥ k 0 , the system energy state can be driven from E(k 0 ) = E 0 ∈ R q + to E(K) = 0byS(·) ∈ ᐁ c .Furthermore,letᐁ be an input space that is a subset of bounded continuous R q -valued functions on Z. The spaces ᐁ r , ᐁ c ,andᐁ are assumed to be closed under the shift operator; that is, if S(·) ∈ ᐁ (resp., ᐁ c or ᐁ r ), then the function S K defined by S K (k) = S(k +K) is contained in ᐁ (resp., ᐁ c or ᐁ r )forallK ≥ 0. 3.2. Nonconservation of entropy and the second law of thermodynamics. The non- linear energy balance equation (3.2) can exhibit a full range of nonlinear behavior in- cluding bifurcations, limit cycles, and even chaos. However, a thermodynamically consis- tent energy flow model should ensure that the evolution of the system energy is diffusive (parabolic) in character with convergent subsystem energies. Hence, to ensure a ther- modynamically consistent energy flow model, we require the following axioms. For the statement of these axioms, we first recall the following graph-theoretic notions. Definit ion 3.2 [2]. A directed graph G(Ꮿ) associated with the connectivity matrix Ꮿ ∈ R q×q has vertices {1, 2, , q} and an arc from vertex i to vertex j, i = j,ifandonlyifᏯ (j,i) = 0. A graph G(Ꮿ) associated with the connectivit y matrix Ꮿ ∈ R q×q is a directed g raph for which the arc s et is symmetric; that is, Ꮿ = Ꮿ T .ItissaidthatG(Ꮿ)isstrongly connected if for any ordered pair of vertices (i, j), i = j, there exists a path (i.e., sequence of arcs) leading from i to j. Recall that Ꮿ ∈ R q×q is irreducible; that is, there does not exist a permutation matrix such that Ꮿ is cogredient to a lower-block triangular matrix, if a nd only if G(Ꮿ)isstrongly connected (see [2, Theorem 2.7]). Let φ ij (E)  σ ij (E) − σ ji (E), E ∈ R q + , denote the net energy exchange between subsystems Ᏻ i and Ᏻ j of the discrete-time large-scale dynamical system Ᏻ. Axiom 1. For the connectivit y matr ix Ꮿ ∈ R q×q associated with the large-scale dynamical system Ᏻ defined by Ꮿ (i, j) =    0 if φ ij (E) ≡ 0, 1 otherwise, i = j, i, j = 1, , q, Ꮿ (i,i) =− q  k=1, k=i Ꮿ (k,i) , i = j, i = 1, ,q, (3.12) rankᏯ = q − 1,andforᏯ (i, j) = 1, i = j, φ ij (E) = 0 if and only if E i = E j . Wassim M. Haddad et al. 283 Axiom 2. For i, j = 1, , q, (E i − E j )φ ij (E) ≤ 0, E ∈ R q + . Axiom 3. For i, j = 1, , q, (∆E i − ∆E j )/(E i − E j ) ≥−1, E i = E j . The fact that φ ij (E) = 0ifandonlyifE i = E j , i = j, implies that subsystems Ᏻ i and Ᏻ j of Ᏻ are connected;alternatively,φ ij (E) ≡ 0 implies that Ᏻ i and Ᏻ j are disconnected. Axiom 1 implies that if the energies in the connected subsystems Ᏻ i and Ᏻ j are equal, then energy exchange between these subsystems is not possible. T his is a statement consistent with the zeroth law of thermodynamics which postulates that temperature equality is a necessary and sufficient condition for thermal equilibrium. Furthermore, it follows from the fact that Ꮿ = Ꮿ T and rankᏯ = q − 1 that the connectivity matr ix Ꮿ is irreducible which implies that for any pair of subsystems Ᏻ i and Ᏻ j , i = j,ofᏳ, there exists a sequence of connected subsystems of Ᏻ that connect Ᏻ i and Ᏻ j . Axiom 2 implies that energy is exchanged from more energetic subsystems to less energetic subsystems and is consistent with the second law of thermodynamics which states that heat (energy) must flow in the direction of lower temperatures. Furthermore, note that φ ij (E) =−φ ji (E), E ∈ R q + , i = j, i, j = 1, ,q, which implies conservation of energy between lossless subsystems. With S(k) ≡ 0, Axioms 1 and 2 along with the fact that φ ij (E) =−φ ji (E), E ∈ R q + , i = j, i, j = 1, ,q, imply that at a given instant of time, energy can only be transported, stored, or dissipated but not created and the maximum amount of energy that can be transported and/or dissipated from a subsystem cannot exceed the energy i n the subsystem. Finally, Axiom 3 implies that for any pair of connected subsystems Ᏻ i and Ᏻ j , i = j, the energy difference between consecutive time instants is monotonic; that is, [E i (k +1)− E j (k + 1)][E i (k) − E j (k)] ≥ 0forallE i = E j , k ≥ k 0 , i, j = 1, ,q. Next, we establish a Clausius-type inequality for our t hermodynamically consistent energy flow model. Proposition 3.3. Consider the discrete-time large-scale dynamical system Ᏻ with energy balance equation (3.2) and assume that Axioms 1, 2,and3 hold. Then for all E 0 ∈ R q + , k f ≥ k 0 ,andS(·) ∈ ᐁ such that E(k f ) = E(k 0 ) = E 0 , k f −1  k=k 0 q  i=1 S i (k) − σ ii  E(k)  c + E i (k +1) = k f −1  k=k 0 q  i=1 Q i (k) c + E i (k +1) ≤ 0, (3.13) where c>0, Q i (k)  S i (k) − σ ii (E(k)), i = 1, ,q, is the amount of net energy (heat) re- ceived by the ith subsystem at the kth instant, and E(k), k ≥ k 0 , is the solution to (3.2)with initial condition E(k 0 ) = E 0 . Furthermore, equality holds in (3.13)ifandonlyif∆E i (k) = 0, i = 1, ,q,andE i (k) = E j (k), i, j = 1, ,q, i = j, k ∈{k 0 , ,k f − 1}. Proof. Since E(k) ≥≥ 0, k ≥ k 0 ,andφ ij (E) =−φ ji (E), E ∈ R q + , i = j, i, j = 1, ,q,itfol- lows from (3.2), Axioms 2 and 3, and the fact that x/(x +1) ≤ log e (1 + x), x>−1 284 Thermodynamic modeling for discrete-time systems that k f −1  k=k 0 q  i=1 Q i (k) c + E i (k +1) = k f −1  k=k 0 q  i=1 ∆E i (k) −  q j=1, j=i φ ij  E(k)  c + E i (k +1) = k f −1  k=k 0 q  i=1  ∆E i (k) c + E i (k)  1+ ∆E i (k) c + E i (k)  −1 − k f −1  k=k 0 q  i=1 q  j=1, j=i φ ij  E(k)  c + E i (k +1) ≤ q  i=1 log e  c + E i  k f  c + E i  k 0   − k f −1  k=k 0 q  i=1 q  j=1, j=i φ ij  E(k)  c + E i (k +1) =− k f −1  k=k 0 q−1  i=1 q  j=i+1  φ ij  E(k)  c + E i (k +1) − φ ij  E(k)  c + E j (k +1)  =− k f −1  k=k 0 q−1  i=1 q  j=i+1 φ ij  E(k)  E j (k +1)− E i (k +1)   c + E i (k +1)  c + E j (k +1)  ≤ 0, (3.14) which proves (3.13). Alternatively, equality holds in (3.13)ifandonlyif  k f −1 k=k 0 (∆E i (k)/(c + E i (k + 1))) = 0, i = 1, ,q,andφ ij (E(k))(E j (k +1)− E i (k +1))= 0, i, j = 1, , q, i = j, k ≥ k 0 .Moreover,  k f −1 k=k 0 (∆E i (k)/(c + E i (k + 1))) = 0isequivalentto∆E i (k) = 0, i = 1, ,q, k ∈{k 0 , ,k f − 1}.Hence,φ ij (E(k))(E j (k +1)− E i (k +1))= φ ij (E(k))(E j (k) − E i (k)) = 0, i, j = 1, ,q, i = j, k ≥ k 0 . Thus, it follows from Axioms 1, 2,and3 that equality holds in (3.13)ifand only if ∆E i = 0, i = 1, , q,andE j = E i , i, j = 1, ,q, i = j.  Inequality (3.13) is analogous to Clausius’ inequality for reversible and irreversible thermodynamics as applied to discrete-time large-scale dynamical systems. It follows from Axiom 1 and (3.2)thatfortheisolated discrete-time large-scale dynamical system Ᏻ; that is, S(k) ≡ 0andd(E(k)) ≡ 0, the energy states given by E e = αe, α ≥ 0, correspond to the equilibrium energy states of Ᏻ.Thus,wecandefineanequilibrium process as a process where the trajector y of the discrete-time large-scale dynamical system Ᏻ stays at t he equi- librium point of the isolated system Ᏻ. The input that can generate such a trajectory can be given by S(k) = d(E(k)), k ≥ k 0 .Alternatively,anonequilibrium process is a process that is not an equilibrium one. Hence, it follows from Axiom 1 that for an equilibrium pro- cess, φ ij (E(k)) ≡ 0, k ≥ k 0 , i = j, i, j = 1, ,q, and thus, by Proposition 3.3 and ∆E i = 0, i = 1, ,q, inequality (3.13) is satisfied as an equality. Alternatively, for a nonequilibrium process, it follows from Axioms 1, 2,and3 that (3.13) is satisfied as a strict inequality. Next, we give a deterministic definition of entropy for the discrete-time large-scale dynamical system Ᏻ that is consistent with the classical thermodynamic definition of entropy. [...]... over all subsystems (iv) For the system Ᏻ, ᏿(E) ≥ 0 for all E ∈ R+ (v) It follows from Proposition 3.17 that for a given value β ≥ 0 of the total energy of the system Ᏻ, one and only one state; namely, E∗ = (β/q)e, corresponds to the largest value of ᏿(E) (vi) It follows from (3.28) that for the system Ᏻ, graph of entropy versus energy is concave and smooth (vii) For a composite discrete-time large-scale... dynamical system ᏳC of two dynamical systems ᏳA and ᏳB the expression for the composite entropy ᏿C = ᏿A + ᏿B , where ᏿A and ᏿B are entropies of ᏳA and ᏳB , respectively, is such that the expression for the equilibrium state where the composite maximum entropy is achieved is identical to those obtained for ᏳA and ᏳB individually Specifically, if qA and qB denote 302 Thermodynamic modeling for discrete-time. .. equality for an equilibrium process and as a strict inequality for a nonequilibrium process It follows from (3.45) that ectropy is a measure of the extent to which the system energy deviates from a homogeneous state Thus, ectropy is the dual of entropy and is a measure of the tendency of the discrete-time large-scale dynamical system Ᏻ to do useful work and grow more organized 3.4 Semistability of thermodynamic. .. nonconservation of entropy and ectropy in the isolated discrete-time large-scale dynamical system Ᏻ implies Axiom 2 The above inequality Wassim M Haddad et al 301 postulates that the direction of energy exchange for any pair of energy similar subsystems is consistent; that is, if for a given pair of connected subsystems at given different energy levels the energy flows in a certain direction, then for any other... (4.7) q and all ectropy functions Ᏹ(E), E ∈ R+ , for Ᏻ satisfy q Ᏹa (E) ≤ Ᏹ(E) − Ᏹ(0) ≤ Ᏹr (E), E ∈ R+ (4.8) Proof The proof is identical to the proofs of Theorems 3.5 and 3.11 For the statement of the next result, recall the definition of p = [1/β1 , ,1/βq ]T and define P diag[β1 , ,βq ] Proposition 4.5 Consider the discrete-time large-scale dynamical system Ᏻ with energy balance equation (3.2) and assume... system entropy varies by an amount q ∆᏿ E(k) ≥ Qi (k) , c + βi Ei (k + 1) i=1 k ≥ k0 (4.11) Finally, note that the nonconservation of entropy and ectropy equations (3.47) and (3.48), respectively, for isolated discrete-time large-scale dynamical systems also hold for the more general definitions of entropy and ectropy given in Definitions 4.2 and 4.3 The following theorem is a generalization of Theorem... temperature 306 Thermodynamic modeling for discrete-time systems equipartition in which all the system energy is eventually transformed into heat at a uniform temperature, and hence all system motion would cease Proposition 4.8 Consider the discrete-time large-scale dynamical system Ᏻ with energy q q balance equation (3.2), let Ᏹ : R+ → R+ and ᏿ : R+ → R denote the ectropy and entropy of q Ᏻ and be given... temperatures of all subsystems are equal 5 Thermodynamic models with linear energy exchange In this section, we specialize the results of Section 3 to the case of large-scale dynamical systems with linear energy exchange between subsystems; that is, w(E) = WE and d(E) = DE, where W ∈ Rq×q and D ∈ Rq×q In this case, the vector form of the energy balance 308 Thermodynamic modeling for discrete-time. .. with energy balance equation (3.2) and assume that Axioms 2 and 3 hold Then there exist an entropy and q q an ectropy function for Ᏻ Moreover, ᏿a (E), E ∈ R+ , and ᏿r (E), E ∈ R+ , are possible enq q tropy functions for Ᏻ with ᏿a (0) = ᏿r (0) = 0, and Ᏹa (E), E ∈ R+ , and Ᏹr (E), E ∈ R+ , are possible ectropy functions for Ᏻ with Ᏹa (0) = Ᏹr (0) = 0 Finally, all entropy functions ᏿(E), q E ∈ R+ , for. .. in (3.30) if and only if ∆Ei = 0, i = 1, , q, and E j = Ei , i, j = 1, , q, i = j 290 Thermodynamic modeling for discrete-time systems Note that inequality (3.30) is satisfied as an equality for an equilibrium process and as a strict inequality for a nonequilibrium process Next, we present the definition of ectropy for the discrete-time large-scale dynamical system Ᏻ Definition 3.10 For the discrete-time . THERMODYNAMIC MODELING, ENERGY EQUIPARTITION, AND NONCONSERVATION OF ENTROPY FOR DISCRETE-TIME DYNAMICAL SYSTEMS WASSIM M. HADDAD, QING HUI, SERGEY G. NERSESOV, AND VIJAYSEKHAR. norm, ᏾(M )and (M)forthe range space and the null space of a matrix M, respectively, spec(M) for the spectrum of the square mat rix M,rank(M) for the rank of the matrix M,ind(M) for the index of M; that. state- ment of the first law of thermodynamics for t he discrete-time large-scale dynamical sys- tem Ᏻ and gives a precise formulation of the equivalence between variation in system internal energy and