Mechanics and Thermodynamics fundamentally united by density operators with an ontic status obeying a locally

Transcription

1 G.P. Beretta, PIAF '09 "New Perspectives on the Quantum State", Perimeter Institute, Sept.27-Oct.2, 2009 Mechanics and Thermodynamics fundamentally united by density operators with an ontic status obeying a locally maximum entropy production dynamics. But at what price? Gian Paolo Beretta Università di Brescia, Italy z Energy, E/hν Entropy, S/k x h y

2 G.P. Beretta, PIAF '09 "New Perspectives on the Quantum State", Perimeter Institute, Sept.27-Oct.2, 2009 The Keenan school of Thermodynamics from engineering, to physics, to mathematical-physics, and back! Keenan MIT press 1941 Hatsopoulos-Keenan Wiley 1965 Recent Symposium AIP 2008 Gyftopoulos-Beretta Macmillan 1991 (Dover 2005)

3 Outline 1) Concept of individual state and an unambiguous representation of preparations, not based on the (von Neumann) density operator 2) An ansatz (1976) allows to embed Thermodynamics into Quantum Theory Ontic status of density operators and microscopic entropy 3) Geometrical construction (1981) of a 'maximal entropy generation' dynamics Ontic and microscopic status to the second law and to irreversibility 4) The price to pay, due to nonlinearity of the dynamical law G.P. Beretta, PIAF '09 "New Perspectives on the Quantum State", Perimeter Institute, Sept.27-Oct.2, 2009

4 Part I 1) Concept of individual state and an unambiguous representation of preparations, not based on the (von Neumann) density operator 2) An ansatz (1976) allows to embed Thermodynamics into Quantum Theory Ontic status of density operators and microscopic entropy 3) Geometrical construction (1981) of a 'maximal entropy generation' dynamics Ontic and microscopic status to the second law and to irreversibility 4) The price to pay, due to nonlinearity of the dynamical law G.P. Beretta, PIAF '09 "New Perspectives on the Quantum State", Perimeter Institute, Sept.27-Oct.2, 2009

5 Part I 1) Concept of individual state and an unambiguous representation of preparations, not based on the (von Neumann) density operator In 1970's language: On the distinction between quantal and nonquantal uncertainties In today's language: On the operational/instrumental distinction between ontic and epistemic probabilities In 1930's language: On the unambiguous mathematical representation of measurement statistics from homogeneous and heterogeneous ensembles G.P. Beretta, PIAF '09 "New Perspectives on the Quantum State", Perimeter Institute, Sept.27-Oct.2, 2009

6 Tomography of a preparation (or ensemble) at time t Ensemble E of identical systems all prepared by at time t Statistics of measurement outcomes (all at time t) Park and Band, FoundPhys, 1, 133 (1970); 1, 211 (1971); 1, 339 (1971) Measurement procedure A a 1 a 2 a d Preparation of system Measurement procedure B b 1 b 2 b d Measurement procedure Z z 1 z 2 z d G.P. Beretta, PIAF '09 "New Perspectives on the Quantum State", Perimeter Institute, Sept.27-Oct.2, 2009

7 G.P. Beretta, PIAF '09 "New Perspectives on the Quantum State", Perimeter Institute, Sept.27-Oct.2, 2009 Statistical mixing of preparations (or ensembles) Preparation Preparation 1 of system w 1 Preparation 2 of system w 2 Homogeneous vs Heterogeneous preparations (or ensembles)

8 Homogeneous preparations (or ensembles) and states G.P. Beretta, PIAF '09 "New Perspectives on the Quantum State", Perimeter Institute, Sept.27-Oct.2, 2009 Preparation o of system Von Neumann, book, engl.transl.1932 Homogeneous vs Heterogeneous preparations (or ensembles)

9 Homogeneous preparations (or ensembles) and states G.P. Beretta, PIAF '09 "New Perspectives on the Quantum State", Perimeter Institute, Sept.27-Oct.2, 2009 Preparation o of system Von Neumann, book, engl.transl.1932 J.L. Park, AmJPhys., 36, 211 (1968) - "Nature of quantum states"

10 Homogeneous preparations (or ensembles) and states G.P. Beretta, PIAF '09 "New Perspectives on the Quantum State", Perimeter Institute, Sept.27-Oct.2, 2009 Homogeneous preparation 1 o w 1 Homogeneous preparation 3 o w 3 Homogeneous preparation 2 o w 2 Homogeneous preparation 4 o w 4 State is either or Contradiction! State is either or Schrödinger, PCPS, 32, 446 (1936) Park&Band, FoundPhys, 6, 157 (1976) quant-ph/ ModPhysLettA, 21, 2799 (2006)

11 Conditions for a mathematical representation compatible with the notion of individual state G.P. Beretta, PIAF '09 "New Perspectives on the Quantum State", Perimeter Institute, Sept.27-Oct.2, 2009 PhD Thesis (1981), quant-ph/

12 Isolated two-level system, standard Quantum Mechanics Spin up Hamiltonian, hω h State, r(t) Energy, E = hω h r(t) x r(t) r& = ω h r i Spin down ψ& = Hψ h i P& ψ = H, Pψ Int.J.Theor.Phys., 24, 119 (1985) h G.P. Beretta, Wor kshop on P er spectiv es in Probability Theor y and its Connections with Science and Society Levico Terme (Trento), Italy, December 3-7, References availab le at: thermodynamics.org h z Bloch sphere, r = r = 1 y 1 Isolated and uncorrelated 2-level particle Spin up Hamiltonian, hω h State, r(t) Energy, E = hω h r(t) z h Bloch sphere, r = r =1 Energy, E/ħω 0-1 Spin down x r& = ω h r i ψ& = Hψ h i P& ψ = h [ H, P ] ψ r(t) G.P. Beretta, PIAF '09 "New Perspectives on the Quantum State", Perimeter Institute, Sept.27-Oct.2, 2009 Energy, E/?ω [ ] y

13 G.P. Beretta, PIAF '09 "New Perspectives on the Quantum State", Perimeter Institute, Sept.27-Oct.2, 2009 The von Neumann representation of heterogeneous z preparations via density operators is incompatible with the notion of individual state ψ ψ Schrödinger, PCPS, 32, 446 (1936) ParkBand-FoundPhys many others later quant-ph/ ModPhysLettA ρ φ φ x y

14 A representation of heterogeneous preparations via probability density distributions over the set of possible ontic states z x y Is this a fundamental indication (theorem) there must exist some measurable observables which need to be References represented available at: by nonlinear functionals of ψ ψ? G.P. Beretta, PIAF '09 "New Perspectives on the Quantum State", Perimeter Institute, Sept.27-Oct.2, 2009

15 Part II 1) Concept of individual state and an unambiguous representation of preparations, not based on the (von Neumann) density operator 2) An ansatz (1976) allows to embed Thermodynamics into Quantum Theory Ontic status of density operators and microscopic entropy 3) Geometrical construction (1981) of a 'maximal entropy generation' dynamics Ontic and microscopic status to the second law and to irreversibility 4) The price to pay, due to nonlinearity of the dynamical law G.P. Beretta, PIAF '09 "New Perspectives on the Quantum State", Perimeter Institute, Sept.27-Oct.2, 2009

16 Part II 1) Concept of individual state and an unambiguous representation of preparations, not based on the (von Neumann) density operator 2) An ansatz (1976) allows to embed Thermodynamics into Quantum Theory Ontic status of density operators and microscopic entropy The need to represent some measurable observables by nonlinear functionals comes from thermodynamics (at least in our engineering "very ontic" view!) Adiabatic availability of nonequilibrium states Entropy Spontaneous relaxation towards equilibrium G.P. Beretta, PIAF '09 "New Perspectives on the Quantum State", Perimeter Institute, Sept.27-Oct.2, 2009

17 G.P. Beretta, PIAF '09 "New Perspectives on the Quantum State", Perimeter Institute, Sept.27-Oct.2, 2009 A physical ansatz (departs drastically from Stat.Mech.) sufficient to unite Mechanics and Thermodynamics The density matrix ρ, even if non-pure, represents a real ontological object, the actual state of the world (system, single particle, even if unentangled). ρ is not understood as needed to represent an epistemic ignorance of which particular pure state the world is really in. The real state is ρ. GN Hatsopoulos and EP Gyftopoulos, Found.Phys., Vol. 6, 15, 127, 439, 561 (1976) The ontic status attributed to the density matrix also legitimates treating the entropy k B Tr ρ lnρ as an ontic physical quantity, like energy or mass. k B Tr ρ lnρ is not understood as measuring how broad is an epistemic probability distribution.

18 1 Isolated and uncorrelated 2-level particle Hamiltonian, hω h State, r(t) Energy, E = hω h r(t) z h Ontic states also inside the Bloch ball r = r <1 Energy, E/ħω 0 x r(t) y ,5 Entropy, S Isoentropic surfaces S = k 1 + r 1+ r 1 r 1 ln + ln r B G.P. Beretta, PIAF '09 "New Perspectives on the Quantum 2State", Perimeter 2 Institute, 2 Sept.27-Oct.2,

19 1 Isolated and uncorrelated 2-level particle Hamiltonian, hω h State, r(t) Energy, E = h ω h r(t ) z h Ontic states also inside the Bloch ball r = r <1 Energy, E/ħω 0 x y ,5 Entropy, S Isoentropic surfaces Max S for given E = 1 r r r S k ln + ln r B G.P. Beretta, PIAF '09 "New Perspectives on the Quantum 2State", Perimeter 2 Institute, 2 Sept.27-Oct.2,

20 Isolated two-level system, Quantum Thermodynamics 1 0 Hamiltonian, hω h State, r(t) Energy, E = h ω h r(t ) x ,5 Entropy, S S 0 Int.J.Theor.Phys., 24, 119 (1985) G.P. Beretta, Wor kshop on Per spectives in Pr obability Theory and its Connections with Science and Society Levico Terme (Trento), Italy, December 3-7, h z Inside the Bloch sphere, r = r <1 y 1 Isolated and uncorrelated 2-level particle Hamiltonian, hω h State, r(t) Energy, E = h ω h r(t ) z h Ontic states also inside the Bloch ball r = r <1 Energy, E/ħω 0 x y ,5 Entropy, S Int.J.Theor.Phys., 24, 119 (1985) f (r) r & = ω h r h r h τ 0 G.P. Beretta, PIAF '09 "New Perspectives on the Quantum State", Perimeter Institute, Sept.27-Oct.2, 2009 S Energy, E/?ω

21 Isolated two-level system, Quantum Thermodynamics Hamiltonian, hω h On the surface of z State, r(t) the Bloch sphere, 1 Energy, E = h ω h r(t ) r = r = 1 h 0 x y r& = ω h r i -1 ψ& = Hψ h 0 0,5 Entropy, S i P& ψ = H, P S = 0 ψ Int.J.Theor.Phys., 24, 119 (1985) h G.P. Beretta, Wor kshop on P er spectiv es in Probability Theor y and its Connections with Science and Society Levico Terme (Trento), Italy, December 3-7, References availab le at: thermodynamics.org 1 Isolated and uncorrelated 2-level particle Hamiltonian, hω h State, r(t) Energy, E = h ω h r(t ) z h On the surface of the Bloch sphere, r = r =1 Energy, E/ħω ,5 Entropy, S Int.J.Theor.Phys., 24, 119 (1985) x r& = ω h r i ψ& = Hψ h i P& ψ = h [ H, P ] ψ = 0 G.P. Beretta, PIAF '09 "New Perspectives on the Quantum State", Perimeter Institute, Sept.27-Oct.2, 2009 S Energy, E/?ω [ ] y

22 Ontic interpretation of the eigenvalues of ρ G.P. Beretta, PIAF '09 "New Perspectives on the Quantum State", Perimeter Institute, Sept.27-Oct.2, 2009 Energy levels E The eigenvalues usually interpreted as "probabilities" can be physically thought as measuring the degree of involvement of S Isolated and uncorrelated N-level particle = = k j B p j e global degree of j j p j e j energy (assuming ρh ln p j level p j j = 1,2,..., N of e j (eigenvalues of = Hρ) the density operator ρ in sharing the energy load entropy measures the H) sharing the energy load among levels / 2 3 / 2 5 / 2 7 / 2 9 / 2 11/ 2 13 / 2 1/ 2 3 / 2 5 / 2 7 / 2 9 / 2 11/ 2 13 / 2 1/ 2 3 / 2 5 / 2 7 / 2 9 / 2 11/ 2 13 / 2 1/ 2 3 / 2 5 / 2 7 / 2 9 / 2 11/ 2 13 / 2 1/ 2 3 / 2 5 / 2 7 / 2 9 / 2 11/ 2 13 / 2 Example, N=7, different distributions with same E and S

23 G.P. Beretta, PIAF '09 "New Perspectives on the Quantum State", Perimeter Institute, Sept.27-Oct.2, 2009 p Giving an "ontic" status to entropy and the Second Law j E = S eigenvalues of ρ (for simplicity, assume ρh = Hρ) = k j B p j e j j p j energy ln p j entropy : global degree of sharing 6.5 In Quantum Mechanics, an isolated and uncorrelated particle is always thought as being in a pure state. If ρh=hρ, this means that only one energy level carries the energy Energy, E/hν /2 3/2 5/2 7/2 9/2 11/2 13/2

24 G.P. Beretta, PIAF '09 "New Perspectives on the Quantum State", Perimeter Institute, Sept.27-Oct.2, 2009 p E S Giving an "ontic" status to entropy and the Second Law j = = k eigenvalues of j B p j e j j p j energy ln p j ρ (for simplicity, assume ρh = Hρ) entropy : global degree of sharing In equilibrium Quantum Thermodynamics, the only stable distribution for the given E, is that which maximizes S In Quantum Mechanics, an isolated and uncorrelated particle is always thought as being in a pure state. If ρh=hρ, this means that only one energy level carries the energy Energy, E/hν /2 3/2 5/2 7/2 9/2 11/2 13/2 1/ 1/ /2 5 5/2 / 2 7 7/2 / 2 9 9/2 / 2 11/2 2 13/2 exp max S ( ) max S (012...inf) ( e k T ) j B 1.5 p j = Entropy, S/k exp ei kb i ( T ) canonical distribution

25 G.P. Beretta, PIAF '09 "New Perspectives on the Quantum State", Perimeter Institute, Sept.27-Oct.2, 2009 p E S Giving an "ontic" status to entropy and the Second Law j = = k eigenvalues of j B p j e j j p j energy ln p j ρ In Quantum Mechanics, an isolated and uncorrelated particle is always thought as being in a pure state. If ρh=hρ, this means that only one energy level carries the energy (for simplicity, assume ρh = Hρ) entropy : global degree of Energy, E/hν / 1/ /2 5 5/2 / 2 7 7/2 / 2 9 9/2 / 2 11/2 2 13/2 1 0 sharing /2 3/20.85/2 7/2 9/2 11/2 13/ /2 3/20.85/2 7/2 9/2 11/2 13/ /2 3/2 5/2 7/2 9/2 11/2 13/ /2 3/2 5/2 7/2 9/2 11/2 13/2 1/2 3/2 5/2 7/2 9/2 11/2 13/ / 2 3 / 2 5 / 2 7 / 2 9 /2 11/2 13 / 2 In equilibrium Quantum Thermodynamics, the only stable distribution for the given E, is that which maximizes S exp max S ( ) max S (012...inf) ( e k T ) j B 1.5 p j = Entropy, S/k exp ei kb i All other distributions with the given E, cannot be stable equilibrium ( T ) canonical distribution

26 Giving an "ontic" status to irreversibility G.P. Beretta, PIAF '09 "New Perspectives on the Quantum State", Perimeter Institute, Sept.27-Oct.2, 2009 Energia, E/hΠ E/hν /2 3/2 5/2 7/2 9/2 11/2 13/2 max S ( ) How to construct an equation of motion for ρ which has the canonical equilibrium distribution as the only stable equilibrium one for each given value of E? max S ( ) Entropia, S/k In Quantum Thermodynamics, irreversible time evolution, for ρh=hρ, can be interpreted as the internal redistribution of energy among all the eigenmodes of energy storage that are acecssible to the particle Energia, E/h E/hν / 2 3 / 2 5/ 2 7/2 9 / 2 11/ 2 13 / Entropia, S/k

27 Unitary dynamics cannot decribe relaxation to equilibrium, nor extraction of the adiabatic availability G.P. Beretta, PIAF '09 "New Perspectives on the Quantum State", Perimeter Institute, Sept.27-Oct.2, 2009 Energy, E/hν nonequilibrium state extraction of the adiabatic availability relaxation Entropy, S/k max S ( ) max S (012...inf) because the eigenvalues of the density operator change with time in these processes.

28 No need for linearity of the dynamical law G.P. Beretta, PIAF '09 "New Perspectives on the Quantum State", Perimeter Institute, Sept.27-Oct.2, 2009 Homogeneous preparation 1 o w 1 Homogeneous preparation 2 o w 2 State at time t is either or Linearity is instead required by the (von Neumann) epistemic view

29 Part III 1) Concept of individual state and an unambiguous representation of preparations, not based on the (von Neumann) density operator 2) An ansatz (1976) allows to embed Thermodynamics into Quantum Theory Ontic status of density operators and microscopic entropy 3) Geometrical construction (1981) of a 'maximal entropy generation' dynamics Ontic and microscopic status to the second law and to irreversibility 4) The price to pay, due to nonlinearity of the dynamical law G.P. Beretta, PIAF '09 "New Perspectives on the Quantum State", Perimeter Institute, Sept.27-Oct.2, 2009

30 Part III 1) Concept of individual state and an unambiguous representation of preparations, not based on the (von Neumann) density operator 2) An ansatz (1976) allows to embed Thermodynamics into Quantum Theory Ontic status of density operators and microscopic entropy 3) Geometrical construction (1981) of a 'maximal entropy generation' dynamics Ontic and microscopic status to the second law and to irreversibility "Design of the equation" using square-root density operator G.P. Beretta, PIAF '09 "New Perspectives on the Quantum State", Perimeter Institute, Sept.27-Oct.2, 2009

31 G.P. Beretta, PIAF '09 "New Perspectives on the Quantum State", Perimeter Institute, Sept.27-Oct.2, 2009 Steepest-entropy-ascent: why square-root probabilities γ(t) γ 1 γ 2 Wootters, Phys.Rev.D, 23, 357 (1981)

32 G.P. Beretta, PIAF '09 "New Perspectives on the Quantum State", Perimeter Institute, Sept.27-Oct.2, 2009 Steepest-entropy-ascent: square root of density operator γ(t) γ 1 γ 2 Ph.D.thesis, MIT (1981) (now at quant-ph/ ) Nuovo Cimento B, 82, 169 (1984) Nuovo Cimento B, 87, 77 (1985) NatoAsi LecNotes, 278, 441 (1986) quant-ph Gheorghiu-Svirschevski, Phys.Rev.A, (2001) quant-ph

33 G.P. Beretta, PIAF '09 "New Perspectives on the Quantum State", Perimeter Institute, Sept.27-Oct.2, 2009 Steepest-entropy-ascent: square root of density operator γ(t) γ 1 γ 2 Ph.D.thesis, MIT (1981) (now at quant-ph/ ) Nuovo Cimento B, 82, 169 (1984) Nuovo Cimento B, 87, 77 (1985) NatoAsi LecNotes, 278, 441 (1986) quant-ph Gheorghiu-Svirschevski, Phys.Rev.A, (2001) quant-ph

34 G.P. Beretta, PIAF '09 "New Perspectives on the Quantum State", Perimeter Institute, Sept.27-Oct.2, 2009 Hamiltonian dynamics: Schroedinger-von Neumann eq. γ& H H γ γ& H S quant-ph/ quant-ph/ quant-ph/ Phys.Rev.E, 73, (2006) Entropy, 10, 160 (2008) Rep.Math.Phys. (2009) quant-ph/

35 G.P. Beretta, PIAF '09 "New Perspectives on the Quantum State", Perimeter Institute, Sept.27-Oct.2, 2009 Hamiltonian dynamics: Schroedinger-von Neumann eq. γ& H H γ γ& H S quant-ph/ quant-ph/ quant-ph/ Phys.Rev.E, 73, (2006) Entropy, 10, 160 (2008) Rep.Math.Phys. (2009) quant-ph/ Exact time-energy uncertainty relation

36 G.P. Beretta, PIAF '09 "New Perspectives on the Quantum State", Perimeter Institute, Sept.27-Oct.2, 2009 Steepest-entropy-ascent: non-hamiltonian contribution γ& D H γ γ& D γ& H quant-ph/ quant-ph/ quant-ph/ Phys.Rev.E, 73, (2006) Entropy, 10, 160 (2008) Rep.Math.Phys. (2009) quant-ph/

37 G.P. Beretta, PIAF '09 "New Perspectives on the Quantum State", Perimeter Institute, Sept.27-Oct.2, 2009 Steepest-entropy-ascent: finding the direction S S L S L H γ quant-ph/ quant-ph/ quant-ph/ Phys.Rev.E, 73, (2006) Entropy, 10, 160 (2008) Rep.Math.Phys. (2009) quant-ph/

38 G.P. Beretta, PIAF '09 "New Perspectives on the Quantum State", Perimeter Institute, Sept.27-Oct.2, 2009 Steepest-entropy-ascent: finding the direction S S L S L H γ quant-ph/ quant-ph/ quant-ph/ Phys.Rev.E, 73, (2006) Entropy, 10, 160 (2008) Rep.Math.Phys. (2009) quant-ph/

39 G.P. Beretta, PIAF '09 "New Perspectives on the Quantum State", Perimeter Institute, Sept.27-Oct.2, 2009 Steepest-entropy-ascent: dynamical law (single isolated particle) Trajectories will be geodesics in the constant energy surface in squareroot density operator space quant-ph/ quant-ph/ quant-ph/ Phys.Rev.E, 73, (2006) Entropy, 10, 160 (2008) Rep.Math.Phys. (2009) quant-ph/

40 G.P. Beretta, PIAF '09 "New Perspectives on the Quantum State", Perimeter Institute, Sept.27-Oct.2, 2009 Steepest-entropy-ascent: dynamical law (single isolated particle) Trajectories will be geodesics in the constant energy surface in squareroot density operator space quant-ph/ quant-ph/ quant-ph/ Phys.Rev.E, 73, (2006) Entropy, 10, 160 (2008) Rep.Math.Phys. (2009) quant-ph/

41 G.P. Beretta, PIAF '09 "New Perspectives on the Quantum State", Perimeter Institute, Sept.27-Oct.2, 2009 Steepest-entropy-ascent: dynamical law (single isolated particle) Trajectories will be geodesics in the constant energy surface in squareroot density operator space quant-ph/ quant-ph/ quant-ph/ Phys.Rev.E, 73, (2006) Entropy, 10, 160 (2008) Rep.Math.Phys. (2009) quant-ph/

42 G.P. Beretta, PIAF '09 "New Perspectives on the Quantum State", Perimeter Institute, Sept.27-Oct.2, 2009 Steepest-entropy-ascent: dynamical law (single isolated particle) τ = intrinsic time characteristic of the relaxation and spontaneous internal redistribution (functional of ρ) quant-ph/ quant-ph/ quant-ph/ Phys.Rev.E, 73, (2006) Entropy, 10, 160 (2008) Rep.Math.Phys. (2009) quant-ph/

43 Part III 1) Concept of individual state and an unambiguous representation of preparations, not based on the (von Neumann) density operator 2) An ansatz (1976) allows to embed Thermodynamics into Quantum Theory Ontic status of density operators and microscopic entropy 3) Geometrical construction (1981) of a 'maximal entropy generation' dynamics Ontic and microscopic status to the second law and to irreversibility Main features of this largely irreversible nonlinear quantum dynamics G.P. Beretta, PIAF '09 "New Perspectives on the Quantum State", Perimeter Institute, Sept.27-Oct.2, 2009

44 G.P. Beretta, PIAF '09 "New Perspectives on the Quantum State", Perimeter Institute, Sept.27-Oct.2, 2009 Steepest-entropy-ascent: rate of entropy increase quant-ph/ quant-ph/ quant-ph/ Phys.Rev.E, 73, (2006) Entropy, 10, 160 (2008) Rep.Math.Phys. (2009) quant-ph/

45 G.P. Beretta, PIAF '09 "New Perspectives on the Quantum State", Perimeter Institute, Sept.27-Oct.2, 2009 Steepest-entropy-ascent: simple form when [H,ρ]=0 Ph.D.thesis, MIT (1981), quant-ph/ Nuovo Cimento B, 82, 169 (1984); 87, 77 (1985) NATO-ASI Lecture Notes, 278, 441 (1986) Phys.Rev.E, 73, (2006) Smooth, constant energy, spontanueous internal redistribution of the eigenvalues p j of ρ, which can be interpreted as degree of energy sharing involvement" of the active levels e j (active means p j > 0).

46 Existence and uniqueness of solutions ρ(t) for any ρ(0) Nuovo Cimento B, 82, 169 (1984); 87, 77 (1985) G.P. Beretta, PIAF '09 "New Perspectives on the Quantum State", Perimeter Institute, Sept.27-Oct.2, 2009

47 G.P. Beretta, PIAF '09 "New Perspectives on the Quantum State", Perimeter Institute, Sept.27-Oct.2, 2009 Existence and uniqueness of solutions ρ(t) for any ρ(0) Dynamical group, not just a semigroup! The inverse map exists Mathematical reversibility

48 G.P. Beretta, PIAF '09 "New Perspectives on the Quantum State", Perimeter Institute, Sept.27-Oct.2, 2009 Entropy increase, metric, and time-entropy uncertainty γ(t) γ 1 γ 2 quant-ph/ quant-ph/ quant-ph/ Phys.Rev.E, 73, (2006) Entropy, 10, 160 (2008) Rep.Math.Phys. (2009) quant-ph/

49 G.P. Beretta, PIAF '09 "New Perspectives on the Quantum State", Perimeter Institute, Sept.27-Oct.2, 2009 Steepest-entropy-ascent: Onsager reciprocity everywhere Found.Phys., 17, 365 (1987) quant-ph Rep.Math.Phys. (2009) quant-ph/

50 G.P. Beretta, PIAF '09 "New Perspectives on the Quantum State", Perimeter Institute, Sept.27-Oct.2, 2009 Steepest-entropy-ascent: Onsager reciprocity everywhere Linear interrelations between rates and affinities. But the L ij s are nonlinear functionals of ρ Found.Phys., 17, 365 (1987) quant-ph Rep.Math.Phys. (2009) quant-ph/

51 Part III 1) Concept of individual state and an unambiguous representation of preparations, not based on the (von Neumann) density operator 2) An ansatz (1976) allows to embed Thermodynamics into Quantum Theory Ontic status of density operators and microscopic entropy 3) Geometrical construction (1981) of a 'maximal entropy generation' dynamics Ontic and microscopic status to the second law and to irreversibility Reduces to the Schroedinger equation for pure density operators The theory is a generalization of Quantum Mechanics which does not contradict it in any way The second law as a theorem about the stability of the equilibrium states G.P. Beretta, PIAF '09 "New Perspectives on the Quantum State", Perimeter Institute, Sept.27-Oct.2, 2009

52 G.P. Beretta, PIAF '09 "New Perspectives on the Quantum State", Perimeter Institute, Sept.27-Oct.2, 2009 Steepest-entropy-ascent: equilibrium states, limit cycles....and the Second Law as a theorem! quant-ph quant-ph quant-ph Phys.Rev.E, 73, (2006)

53 Equilibrium states for a 4-level isolated particle Phys.Rev.E, 73, (2006) e 4 = 1 lowest entropy and unstable equilibrium states e 3 = 2/3 e 2 = 1/3 Energy, E (canonical distribution) stable equilirium states e 1 = 0 Entropy G.P. Beretta, PIAF '09 "New Perspectives on the Quantum State", Perimeter Institute, Sept.27-Oct.2, 2009

54 Part III 1) Concept of individual state and an unambiguous representation of preparations, not based on the (von Neumann) density operator 2) An ansatz (1976) allows to embed Thermodynamics into Quantum Theory Ontic status of density operators and microscopic entropy 3) Geometrical construction (1981) of a 'maximal entropy generation' dynamics Ontic and microscopic status to the second law and to irreversibility Dynamical group, not a semi-group G.P. Beretta, PIAF '09 "New Perspectives on the Quantum State", Perimeter Institute, Sept.27-Oct.2, 2009

55 Relaxation to equilibrium for a 4-level isolated particle N = 4 [H,ρ]=0 An arbitrary initial distribution (state) entropy entropy generation Stable equilibrium state (highest entropy) Phys.Rev.E, 73, (2006) time t/τ G.P. Beretta, PIAF '09 "New Perspectives on the Quantum State", Perimeter Institute, Sept.27-Oct.2, 2009

56 Relaxation to equilibrium for a 4-level isolated particle The trajectory passes very close to an unstable equilibrium state An arbitrary initial distribution (state) Strong causality: given any initial state the trajectory is unique and defined for - < t < + We can trace back the lowest entropy ancestral state entropy entropy generation Stable equilibrium state (highest entropy) Phys.Rev.E, 73, (2006) G.P. Beretta, PIAF '09 "New Perspectives on the Quantum State", time Perimeter t/τ Institute, Sept.27-Oct.2, 2009

57 Relaxation to equilibrium for a 4-level isolated particle time-entropy uncertainty bound entropy entropy generation Stable equilibrium state (highest entropy) Phys.Rev.E, 73, (2006) G.P. Beretta, PIAF '09 "New Perspectives on the Quantum State", time Perimeter t/τ Institute, Sept.27-Oct.2, 2009

58 Part IV 1) Concept of individual state and an unambiguous representation of preparations, not based on the (von Neumann) density operator 2) An ansatz (1976) allows to embed Thermodynamics into Quantum Theory Ontic status of density operators and microscopic entropy 3) Geometrical construction (1981) of a 'maximal entropy generation' dynamics Ontic and microscopic status to the second law and to irreversibility 4) The price to pay, due to nonlinearity of the dynamical law Structure of the nonlinear dynamical law for composite systems Locality and separability G.P. Beretta, PIAF '09 "New Perspectives on the Quantum State", Perimeter Institute, Sept.27-Oct.2, 2009

59 G.P. Beretta, PIAF '09 "New Perspectives on the Quantum State", Perimeter Institute, Sept.27-Oct.2, 2009 No free lunch!* Dealing with composite systems (*Except at PI) A = Alice B = Bob Nuovo Cimento B, 87, 77 (1985) quant-ph Rep.Math.Phys. (2009)

60 G.P. Beretta, PIAF '09 "New Perspectives on the Quantum State", Perimeter Institute, Sept.27-Oct.2, 2009 Steepest-locally-perceived-entropy-ascent Nuovo Cimento B, 87, 77 (1985) quant-ph Rep.Math.Phys. (2009)

61 G.P. Beretta, PIAF '09 "New Perspectives on the Quantum State", Perimeter Institute, Sept.27-Oct.2, 2009 Steepest-locally-perceived-entropy-ascent Nuovo Cimento B, 87, 77 (1985) quant-ph Rep.Math.Phys. (2009)

62 Conclusions 1) Concept of individual state and an unambiguous representation of preparations, not based on the (von Neumann) density operator 2) An ansatz (1976) allows to embed Thermodynamics into Quantum Theory Ontic status of density operators and microscopic entropy 3) Geometrical construction (1981) of a 'maximal entropy generation' dynamics Ontic and microscopic status to the second law and to irreversibility 4) The price to pay, due to nonlinearity of the dynamical law Rediscoveries Epilogue Measurable effects? Summary G.P. Beretta, PIAF '09 "New Perspectives on the Quantum State", Perimeter Institute, Sept.27-Oct.2, 2009

63 Steepest entropy ascent: rediscoveries (15 years later) G.P. Beretta, PIAF '09 "New Perspectives on the Quantum State", Perimeter Institute, Sept.27-Oct.2, 2009

64 G.P. Beretta, PIAF '09 "New Perspectives on the Quantum State", Perimeter Institute, Sept.27-Oct.2, 2009

65 Measurable effects? atomic beam (diluted) (near stable eql.) resonance fluorescence detector laser beam ( pump ) off resonance (detuned) Irreversible internal redistribution implies asymmetries in the spectral distribution Int.J.Theor.Phys., 24, 1233 (1985) G.P. Beretta, PIAF '09 "New Perspectives on the Quantum State", Perimeter Institute, Sept.27-Oct.2, 2009

66 Measurable effects? laser beam ( probe ) atomic beam (diluted) (near stable eql.) laser beam ( pump ) on resonance (tuned) absorption and stimulated emission detector Irreversible internal redistribution implies attenuation of stimulated emission Int.J.Theor.Phys., 24, 1233 (1985) Negative values = stimulated emission G.P. Beretta, PIAF '09 "New Perspectives on the Quantum State", Perimeter Institute, Sept.27-Oct.2, 2009

67 Conclusions - an ontic role of the density operator makes Quantum Thermodynamics fundamental The second law is forced directly into the microscopic laws and emerges as a theorem of the constrained steepest-entropy-ascent (or maximal entropy generation) dynamical law (stability of equilibrium states) Entropy emerges from the microscopic level as a measure of the degree of load sharing among the particle s energy levels. Irreversibility emerges as a manifestation of spontaneous internal load redistribution among the initially occupied energy levels. Yet the dynamical equation is mathematically reversible. Nonlinearity requires a non-universal formal structure of the dynamical law. The structure is "model dependent": like the Hamiltonian, it depends on what subsystems are assumed as elementary and separable, i.e., non communicating. Standard (pure state) quantum mechanics emerges as the (mildly unstable) boundary solutions (limit cycles) of the more general theory The theory is conceptually controversial, but mathematically robust, awaits experimental validation and philosophycal scrutiny G.P. Beretta, PIAF '09 "New Perspectives on the Quantum State", Perimeter Institute, Sept.27-Oct.2, 2009