Non equilibrium thermodynamics: foundations, scope, and extension to the meso scale. Miguel Rubi

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1 Non equilibrium thermodynamics: foundations, scope, and extension to the meso scale Miguel Rubi

2 References S.R. de Groot and P. Mazur, Non equilibrium Thermodynamics, Dover, New York, 1984 J.M. Vilar and J.M. Rubi, PNAS (2001) D. Reguera, J.M. Rubi and J.M. Vilar, J. Phys. Chem. B (2005); Feature Article J.M. Rubi, Scientific American, November (2008)

3 Thermodynamics Heat, work and energy in large scale systems Thermodynamic limit: N V It only characterizes equilibrium states 1 2 process N ρ = V finite States are represented by a reduced number of quantities. Gas: P, V, T; (related through eq of state) Time plays no role; states are quiescent

4 Why a non equilibrium thermodynamics? Thermodynamics describes changes in systems that are in equilibrium S(U,V) TdS = du + pdv μdn S a b a. b U V How to deal with systems outside equilibrium?

5 Non equilibrium systems Properties are not uniform: T(x,t); presence of gradients Systems evolve in time How to describe the evolution? In thermodynamics the evolution is dictated by the second law How the second law applies? Task of NET

6 Non equilibrium thermodynamics Local equilibrium: the system is globally outside equilibrium but is locally in equilibrium. Consequence: thermodynamics is valid locally; the small volume elements are thermodynamics systems. Requirement: the small volume elements must contain enough particles; the system can be treated as a continuum medium.

7 Continuum hypothesis ΔM ΔV I: large fluctuations II: deterministic regime III: inhomogeneities I II III ΔV ρ = lim Δ V 0 ΔM ΔV If fluctuations are negligible then region I is very small and we can consistently define ρ = dm dv

8 Temperature gradient T+n(a/N) T T+a Size L : volume elements; small thermodynamic systems Gradient: a/l Temperature differences: a/n

9 From devices to industrial plants NET: universal Reason: system continuum medium Scaling up

10 An example: Hydrodynamics Scaling of the Navier Stokes equation v 2 + v v = p + υ v t 1,, h h trv λ t, λr, λ v 2h 1 v 1 h 2 2 λ + v v = λ p+ λ υ v t h = 1 h UL λ UλL UL Re = = invariant! υ υ υ

11 Scales Macroscopic: objects Microscopic: molecules Mesoscopic: meso structures (clusters, aggregates, proteins, DNA, )

12 Protein Atomistic Mesoscopic

13 Diffusion Fick J = D ρ i) Large scales ii) Long times Description in terms of average values ρ t = J

14 NET of diffusion Tds = μdρ Gibbs; local equilibrium σ L μ J = ; D= L/ T T x = 1 J T μ x ρ ρ = D t x x

15 Ion pumping Active transport Onsager relation: l rd = l dr lrr lrd o i J = ΔG Ca Ca T T μ μ l J = ΔG l μ μ T o dr o i Ca dd Ca Ca

18 Efficiency (ies) First law efficiency: η = I W Q+ W Second law efficiency: η = II W W ideal ( σ ) Wlost = W Wideal = TΔ Stot = T dv Δt Ex. Aluminium electrolysis: ηi = 0.50 η = 0.42 η = 0.60 II Fuel cells: II

19 From devices to nanosystems Scaling down At short scales matter cannot be considered as a continuum

20 Assumption: fluid is a continuum; ignoring molecular scales Langevin forces: not considered by NET

21 Langevin force t 1 t 2 Random walk Δx < ( Δ x) 2 >= 2Dt α

22 Langevin equation: dv m = ζ v + F dt i) < F ( t) >= 0 R ii) < F ( t) F ( t') >= 2 ktζδ ( t t') R R R Langevin force Diffusion: molecular nature < E >= 1 2 kt

23 Motion of a particle through a liquid v dv m F dt = 1 dv 1 σ = mv = v F T dt T L v= F T F = ζ v b = L / T = ζ 1 F? 1 TdS de d mv 2 2 = p = dv m = ζ v dt ζ v= v0 exp t m

24 Failure of NET: Chemical reactions σ = 1 JA T J L L = A= ( ) 2 1 T T μ μ Law of mass action linearization μ2 μ1 A L kt kt kt J = D e e = D(1 e ) A T Conclusion: NET only accounts for the linear regime

25 Activation over a single barrier Basic relaxation mechanism: Overcoming of a single barrier (Free) Energy Reaction coordinate Q

26 Is thermodynamics applicable to small scale systems outside How small is small? equilibrium? Extensivity validates thermodynamics What is the actual meaning of being far away from equilibrium? Mesoscale local equilibrium validates a mesoscopic nonequilibrium thermodynamics

27 Small scale systems Peculiar features: 1. Free energy contains more contributions G = Nμ ( T, P) + N h( T, P) 2 3 N ; G Nμ Surface contribution

28 Snapshots of the proteins myoglobin (controls the oxigen transport) GFP Ca-ATPase 1SU4 (controls the active transport) E1 Catalytic cycle E2 Ca-ATPase 1KJU

29 Checking the extensivity Heat capacity of proteins It is dominated, about 80% of the total by intramolecular contributions (stretching, bending and internal rotations) Hydration contribution about 15% C V N H, G,... N

30 2. Fluctuations are important A=< A>+δ A thermodynamic value fluctuation Macroscopic: continuum A < A > δ A << 1 < A >

31 Temperature relaxation ----: Model exp( t / τ ) T, p are fast variables; related to collisions (many) Density slower; related to rearrangements τ ps T T i f = 350K = 250K

32 Heat transfer model R Thermal conductivity

33 Can we define a temperature? Gaussian distribution Temperature Myoglobin

34 3. Entropy; Boltzmann versus Gibbs Δ S ( α ) = k ln P( α ) B B Δ = α α P( α, α; t) 0 SG kb P( 0, ; t)ln d P0 α Macroscopic limit: In the macroscopic limit Boltzmann and Gibbs entropies provide the same description <Δ S B > α0 t 0 Second law; macroscopic limit <Δ S > B α0 t 0 <Δ S > G α0 t 0 always

35 Activation Unstable substance Naked eye: Sudden jump Final product Watching closely Diffusion Progressive molecular changes

36 Translocation of ions (through a membrane or protein channel) Biological membrane biological pumps, chemical and biochemical reactions short time scale: local equilibrium along the coordinate Local, linear Global, non linear Arrhenius, Butler Volmer, Law of mass action

37 Activated process viewed as a diffusion process along a reaction coordinate kt kt kt kt ( ) L Φ μ Φ μ μ kl σγ J = = e e De e T γ P γ γ From local to global: γ dγ... dγje = D dγ e γ Φ 2 2 kt 1 1 μ kt μ2 μ1 kt kt J = D e e D ( z2 z1)

38 Molecular changes: diffusion through a mesoscopic coordinate γ : mesoscopic coordinate P( γ, t): probability Second law J.M.Rubi, Scientific American, 299, 62 (2008)

39 Single molecule x:center of mass Θ:size, others Force Local equilibrium: Tds( x) = μ ( x) d ρ ( x) FdΘ ( x) Mesoscale local equilibrium: Θ ( x ) = θ P( x, θ ) dθ Tds( x, θ ) = μ ( x, θ ) dp( x, θ )

40 Meso scale entropy production

41 The noise is the signal Gaussian noise and time-reversal symmetry M Vainstein, JM Rubi; PRE (2007) D = kt γ Thermodynamic force d V ' = kt lnp dx 0 Thermodynamics likes Gaussian stuff

42 RNA unfolding under tension ρ () t = j + j n n n 1 MNET j = k ( n+ 1, f) ρ + k ( n+ 1) ρ n c n+ 1 o n Master equation [ ] ρ ( t) = k ( n+ 1) + k ( n, f) ρ + k ( n) ρ + k ( n+ 1, f) ρ n o c n o n c n J.M. Rubi, D. Bedeaux, S. Kjelstrup, J. Phys. Chem. B (2007) ρ n+ 1, st ρ nst, ( G n f k T) = exp Δ (, ) / B

43 Protein folding Intermediate configurations, same as for chemical reactions

44 Muscular contraction Energy transduction, Molecular motors

45 Heat exchange at the nanoscale Failure of the FDT at small distance d L. Lapas, A. Perez, JMR, PRL (2009)

46 Nucleation Basic scenario: Φ embryo melted crystal Order parameter Metastable phase Cluster at rest : x n Clusterinabath: x ( n, v, ω)

47 Conclusions Many systems we call small still contain many particles and behave thermodynamically NET is not able to describe the activation kinetics but a proper interpretation of the concept of local equilibrium at the mesoscale can do it. This interpretation can be used in chemical and biochemical reactions, transport in confined systems (ion channels, zeolites), single molecules, heat exchange at the nanoscale, nucleation,...

COARSE-GRAINING AND THERMODYNAMICS IN FAR-FROM-EQUILIBRIUM SYSTEMS

Vol. 44 (2013) ACTA PHYSICA POLONICA B No 5 COARSE-GRAINING AND THERMODYNAMICS IN FAR-FROM-EQUILIBRIUM SYSTEMS J. Miguel Rubí, A. Pérez-Madrid Departament de Física Fonamental, Facultat de Física, Universitat