1. The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment. Denis J. Evans, Edie Sevick, Genmaio Wang, David Carberry, Emil Mittag and James Reid Research School of Chemistry, Australian National University, Canberra, Australia and Debra J. Searles Griffith University, Queensland, Australia Other collaborators E.G.D. Cohen, G.P. Morriss, Lamberto Rondoni (March 2006)

2. The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment Fluctuation Theorem (Roughly). The first statement of a Fluctuation Theorem was given by Evans, Cohen & Morriss, 1993. This statement was for isoenergetic nonequilibrium steady states. If is total (extensive) irreversible entropy production rate/ and its time average is:, then Formula is exact if time averages (0,t) begin from the equilibrium phase. It is true asymptotically, if the time averages are taken over steady state trajectory segments. The formula is valid for arbitrary external fields,.

3. The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment Evans, Cohen & Morriss, PRL, 71, 2401(1993).

4. The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment Why are the Fluctuation Theorems important? Show how irreversible macroscopic behaviour arises from time reversible dynamics. Generalize the Second Law of Thermodynamics so that it applies to small systems observed for short times. Implies the Second Law InEquality. Are valid arbitrarily far from equilibrium regime In the linear regime FTs imply both Green-Kubo relations and the Fluctuation dissipation Theorem. Are valid for stochastic systems (Lebowitz & Spohn, Evans & Searles, Crooks). New FT’s can be derived from the Langevin eqn (Reid et al, 2004). A quantum version has been derived (Monnai & Tasaki),. Apply exactly to transient trajectory segments (Evans & Searles 1994) and asymptotically for steady states (Evans et al 1993).. Apply to all types of nonequilibrium system: adiabatic and driven nonequilibrium systems and relaxation to equilibrium (Evans, Searles & Mittag). Can be used to derive nonequilibrium expressions for equilibrium free energy differences (Jarzynski 1997, Crooks). Place (thermodynamic) constraints on the operation of nanomachines.

5. The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment Derivation of TFT (Evans & Searles 1994 - 2002) Consider a system described by the time reversible thermostatted equations of motion (Hoover et al): Example: Sllod NonEquilibrium Molecular Dynamics algorithm for shear viscosity - is exact for adiabatic flows. which is equivalent to: (Evans and Morriss (1984)).

6. The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment The Liouville equation is analogous to the mass continuity equation in fluid mechanics. or for thermostatted systems, as a function of time, along a streamline in phase space:  is called the phase space compression factor and for a system in 3 Cartesian dimensions The formal solution is:

7. The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment Thermostats Deterministic, time reversible, homogeneous thermostats were simultaneously but independently proposed by Hoover and Evans in 1982. Later we realised that the equations of motion could be derived from Gauss' Principle of Least Constraint (Evans, Hoover, Failor, Moran & Ladd (1983)). The form of the equations of motion is  can be chosen such that the energy is constant or such that the kinetic energy is constant. In the latter case the equilibrium, field free distribution function can be proved to be the isokinetic distribution, In 1984 Nosé showed that if  is determined as the time dependent solution of the equation then the equilibrium distribution is canonical

8. The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment Aside: - Thermostats and Equilibrium Consider “  ” thermostats described by the equations of motion: where Einstein notation is used,, is the position of the i-th particle in the  -direction, is the momentum of the ith particle in the  -direction, and couple the system with the external field, At : all  -thermostats that violate Gauss Principle do not generate an equilibrium state and, among  -thermostats that satisfy Gauss's Principle to fix the  +1 moment of the velocity distribution, only the conventional Gaussian isokinetic thermostat (  =1) possesses an equilibrium state.

9. The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment If the total system is Hamiltonian Although  for any Hamiltonian system, in general,  for any subsystem of a Hamiltonian system. Further, if it is easy to show that if a subsystem (ie the system of interest “SOI”) looses heat at a rate to its Hamiltonian surroundings (reservoir RES) and if those surroundings have a heat capacity very much greater than that of the system of interest, so that the surroundings can be regarded as being in thermodynamic equilibrium then, one can show If the reservoir is thermostatted with a  -thermostat with a large number of degrees of freedom, then We see that the phase space compression factor for the system of interest is identical in the two cases. This confirms the fact that the system of interest cannot “know” how the heat is ultimately removed from its vicinity.

10. The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment We know that The dissipation function is in fact a generalized irreversible entropy production - see below. The Dissipation function is defined as:

11. The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment Phase Space and reversibility

12. The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment The Loschmidt Demon applies a time reversal mapping: Loschmidt Demon

13. The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment Combining shows that So we have the Transient Fluctuation Theorem (Evans and Searles 1994) The derivation is complete. Evans Searles TRANSIENT FLUCTUATION THEOREM

14. The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment FT for different ergodically consistent bulk ensembles driven by a dissipative field, Fe with conjugate flux J. Isokinetic or Nose-Hoover dynamics/isokinetic or canonical ensemble Isoenergetic dynamics/microcanonical ensemble or (Note: This second equation is for steady states, the Gallavotti-Cohen form for the FT (1995).) Isobaric-isothermal dynamics and ensemble. (Searles & Evans, J. Chem. Phys., 113, 3503–3509 (2000))

15. The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment Consequences of the FT Connection with Linear irreversible thermodynamics In thermostatted canonical systems where dissipative field is constant, So in the weak field limit (for canonical systems) the average dissipation function is equal to the “rate of spontaneous entropy production” - as appears in linear irreversible thermodynamics. Of course the TFT applies to the nonlinear regime where linear irreversible thermodynamics does not apply.

16. The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment The Integrated Fluctuation Theorem (Ayton, Evans & Searles, 2001). If denotes an average over all fluctuations in which the time integrated entropy production is positive, then, gives the ratio of probabilities that the Second Law will be satisfied rather than violated. The ratio becomes exponentially large with increased time of violation, t, and with system size (since  is extensive).

17. The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment The Second Law Inequality If denotes an average over all fluctuations in which the time integrated entropy production is positive, then, If the pathway is quasi-static (i.e. the system is always in equilibrium): The instantaneous dissipation function may be negative. However its time average cannot be negative. (Searles & Evans 2004).

18. The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment The NonEquilibrium Partition Identity (Carberry et al 2004). For thermostatted systems the NonEquilibrium Partition Identity (NPI) was first proved by Evans & Morriss (1984). It is derived trivially from the TFT. NPI is a necessary but not sufficient condition for the TFT.

19. The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment Steady state Fluctuation Theorem At t=0 we apply a dissipative field to an ensemble of equilibrium systems. We assume that this set of systems comes to a nonequilibrium steady state after a time . For any time t we know that the TFT is valid. Let us approximate, so that Substituting into the TFT gives, In the long time limit we expect a spread of values for typical values of which scale as consequently we expect that for an ensemble of steady state trajectories, (Evans, Searles and Rondoni 2006, Evans & Searles 2000).

20. The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment We expect that if the statistical properties of steady state trajectory segments are independent of the particular equilibrium phase from which they started (the steady state is ergodic over the initial equilibrium states), we can replace the ensemble of steady state trajectories by trajectory segments taken from a single (extremely long) steady state trajectory. This gives the Evans-Searles Steady State Fluctuation Theorem Steady State ESFT

21. The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment FT and Green-Kubo Relations Thermostatted steady state. The SSFT gives Plus Central Limit Theorem Yields in the zero field limit Green-Kubo Relations Note: If t is sufficiently large for SSFT convergence and CLT then is the largest field for which the response can be expected to be linear. (Evans, Searles and Rondoni 2005).

22. The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment NonEquilibrium Free Energy Relations Equilibrium Helmholtz free energy differences can be computed nonequilibrium thermodynamic path integrals. For nonequilibrium isothermal pathways between two equilibrium states implies, NB is the difference in Helmholtz free energies, and if then JE KI Crooks Equality (1999). Jarzynski Equality (1997).

23. The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment Evans, Mol Phys, 20,1551(2003).

24. The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment Connection between FTs Jarzynski and Crooks. For stochastic systems the initial phase does not uniquely determine the “trajectory”, hence the specification of initial and final phases (0,t). Definitions:

25. The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment Crooks proof: systems are deterministic and canonical Jarzynski Equality proof:

26. The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment Then Crooks: Evans_SearlesFT: Reid et al.: and: Reid et.al.: which gives a formal relationship between Crooks (therefore Jarzynski) and Evans and Searles FT.

27. The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment Examples: Microcanonical ensembleCanonical ensemble where and, (Reid et.al. 2005)

28. The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment Jarzynski GK and SLI. If then ie NPI. So if we take the time time weak field limit and assume a finite decay time for correlations, we expect Gaussian statistics. This further implies the FT but there is no need for ergodic consistency.

29. The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment Comments on van Zon & Cohen heat function If then the phase space compression factor and the dissipation function are exactly related by the equation So when van Zon and Cohen introduce the heat function Q, for a single particle obeying the inertialess Langevin equation for a particle in an optical trap And thus when van Zon and Cohen show They show that for colloids whose underlying dynamics is Newtonian, GCFT does not hold.

30. The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment Conflicting views on the Fluctuation Theorem “Sometimes change is a result of an illusory quest for novelty. It is quite possible to pursue blind alleys in physics, roads through an imaginary landscape, which lead nowhere.... So far there is no indication that something like pairing, or a Fluctuation Theorem, holds for a system with realistic nonequilibrium boundary conditions” p236, Time Reversibility, Computer Simulation and Chaos, W. G. Hoover, World Scientific 1999. “The TFT HAS to be satisfied, since it is in a way an identity...I feel that the verification of the TFT is almost more a check on the experiments than on the theorem, because it HAS to hold. Nevertheless it is very nice that you can do this!” E.G.D. Cohen, private correspondence, 30 June 2001.

31. The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment single colloidal particle position & velocity measured precisely impose & measure small forces small system short trajectory small external forces Strategy of experimental demonstration of the FTs... measure energies, to a fraction of, along paths

32. The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment Optical Trap Schematic Photons impart momentum to the particle, directing it towards the most intense part of the beam. r k < 0.1 pN/  m, 1.0 x 10 -5 pN/Å

33. The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment Optical Tweezers Lab quadrant photodiode position detector sensitive to 15 nm, means that we can resolve forces down to 0.001 pN or energy fluctuations of 0.02 pN nm (cf. k B T=4.1 pN nm)

34. The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment As  A=0, and FT and Crooks are equivalent For the drag experiment... velocity time 0 t=0 v opt = 1.25  m/sec Wang, Sevick, Mittag, Searles & Evans, “Experimental Demonstration of Violations of the Second Law of Thermodynamics” Phys. Rev. Lett. (2002)  t > 0, work is required to translate the particle-filled trap  t < 0, heat fluctuations provide useful work “entropy-consuming” trajectory

35. The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment Wang et al PRL, 89, 050601(2002). First demonstration of the (integrated) FT FT shows that entropy-consuming trajectories are observable out to 2-3 seconds in this experiment Wang, Sevick, Mittag, Searles & Evans, Phys. Rev. Lett. (2002)

36. The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment For the Capture experiment... Carberry, Reid, Wang, Sevick, Searles & Evans, Phys. Rev. Lett. (2004) k0k0 k1k1 time t=0 trapping constant k0k0 k1k1

37. The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment Optical Capture of a Brownian Bead. - TFT, NPI For a sudden isothermal change of strength in an optical trap, the dissipation function is: Note: as expected, So the TFT becomes:

38. The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment Histogram of  t for Capture k 0 = 1.22 pN/  m k 1 = (2.90, 2.70) pN/  m predictions from Langevin dynamics Carberry, Reid, Wang, Sevick, Searles & Evans, Phys. Rev. Lett. (2004)

39. The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment The LHS and RHS of the Integrated Transient Fluctuation Theorem (ITFT) versus time, t. Both sets of data were evaluated from 3300 experimental trajectories of a colloidal particle, sampled over a millisecond time interval. We also show a test of the NonEquilibrium Partition Identity. (Carberry et al, PRL, 92, 140601(2004)) ITFT NPI

40. The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment

41. Experimental Tests of Steady State Fluctuation Theorem Colloid particle 6.3 µm in diameter. The optical trapping constant, k, was determined by applying the equipartition theorem: k = k B T/. The trapping constant was determined to be k = 0.12 pN/µm and the relaxation time of the stationary system was  =0.48 s. A single long trajectory was generated by continuously translating the microscope stage in a circular path. The radius of the circular motion was 7.3 µm and the frequency of the circular motion was 4 mHz. The long trajectory was evenly divided into 75 second long, non-overlapping time intervals, then each interval (670 in number) was treated as an independent steady-state trajectory from which we constructed the steady-state dissipation functions.

42. The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment

44. Test of NonEquilibrium Free Energy Theorems for Optical Capture.

45. The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment For the Ramp experiment... trapping constant time t=0 t=  k/k k0k0 k1k1. undefined as the external field is not time-symmetric quasi-static, limit limit is capture

46. The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment Test of NE WR

47. The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment New far-from-equilibrium theorems in statistical physics Crooks Relation Jarzynski Relation Fluctuation Theorem (An extended Second Law-like theorem) NonEquilibrium Partition Identity