Collective Effects. Equilibrium and Nonequilibrium Physics

Transcription

1 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, Collective Effects in Equilibrium and Nonequilibrium Physics Website: Caltech Mirror:

2 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, Today s Lecture Onsager Theory and the Fluctuation-Dissipation Theorem Motivation from lecture 1 Derivation and discussion Application to nanomechanics and biodetectors

3 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, Motivation So far we have only talked about thermodynamic and equilibrium consequences of conservation laws and broken symmetries. In macroscopic systems dissipation is important Dissipation is associated with the increase of entropy, and is outside of the scope of thermodynamics where entropy (or the appropriate thermodynamic potential) is maximized or minimized. Onsager, and later Callan, Greene, Kubo and others showed how to systematically treat systems near equilibrium

4 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, Equilibrium under Energy Transfer J E Energy E 1 Energy E 2 Isolated system divided into two weakly coupled halves or subsystems. Initially the whole system is in thermodynamic equilibrium. Take the system away from equilibrium by transferring an energy E from one half to the other. For weak coupling the time scale for the relaxation will be correspondingly long.

5 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, Dissipative Currents Temperature T 1 Temperature T 2 J E Equilibrium is given by the equality of the temperatures T 1 = T 2. Different temperatures gives a nonequilibrium state, and an energy current moving the system towards equilibrium. For small temperature differences δt = T 2 T 1 J E = K δt K is a kinetic coefficient or dissipation coefficient

6 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, Slow Relaxation The temperatures of the subsystems will change at a rate proportional to the rate of change of energy Ṫ 1 = Ė 1 /C 1 = J E /C 1 Ṫ 2 = Ė 2 /C 2 = J E /C 2 where C i is the thermal capacity of subsystem i.

7 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, Slow Relaxation The temperatures of the subsystems will change at a rate proportional to the rate of change of energy Ṫ 1 = Ė 1 /C 1 = J E /C 1 Ṫ 2 = Ė 2 /C 2 = J E /C 2 where C i is the thermal capacity of subsystem i. Since the relaxation between the systems is slow, each system may be taken as internally in equilibrium, so that C i is the equilibrium value of the specific heat.

8 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, Slow Relaxation The temperatures of the subsystems will change at a rate proportional to the rate of change of energy Ṫ 1 = Ė 1 /C 1 = J E /C 1 Ṫ 2 = Ė 2 /C 2 = J E /C 2 where C i is the thermal capacity of subsystem i. Since the relaxation between the systems is slow, each system may be taken as internally in equilibrium, so that C i is the equilibrium value of the specific heat. Using J E = K δt gives with δṫ = (K /C)δT C 1 = C C 1 2

9 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, Slow Relaxation The temperatures of the subsystems will change at a rate proportional to the rate of change of energy Ṫ 1 = Ė 1 /C 1 = J E /C 1 Ṫ 2 = Ė 2 /C 2 = J E /C 2 where C i is the thermal capacity of subsystem i. Since the relaxation between the systems is slow, each system may be taken as internally in equilibrium, so that C i is the equilibrium value of the specific heat. Using J E = K δt gives with δṫ = (K /C)δT C 1 = C C 1 2 This equation yields exponential relaxation with a time constant given by macroscopic quantities. τ = C/K

10 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, Entropy Production Since the energy current is the process of the approach to equilibrium, the entropy must increase in this relaxation.

11 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, Entropy Production Since the energy current is the process of the approach to equilibrium, the entropy must increase in this relaxation. Rate of change of entropy: Ṡ = Ṡ 1 (E 1 ) + Ṡ 2 (E 2 ) = 1 T 1 Ė T 2 Ė 2 J E δt/t 2 = K (δt/t ) 2

12 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, Entropy Production Since the energy current is the process of the approach to equilibrium, the entropy must increase in this relaxation. Rate of change of entropy: Ṡ = Ṡ 1 (E 1 ) + Ṡ 2 (E 2 ) = 1 T 1 Ė T 2 Ė 2 J E δt/t 2 = K (δt/t ) 2 Second law requires the kinetic coefficient K > 0

13 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, Continuum System: Nonuniform Temperature

14 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, Continuum System: Nonuniform Temperature Macroscopic system is in equilibrium when the temperature is uniform in space.

15 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, Continuum System: Nonuniform Temperature Macroscopic system is in equilibrium when the temperature is uniform in space. Spatially varying temperature will lead to energy flows.

16 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, Continuum System: Nonuniform Temperature Macroscopic system is in equilibrium when the temperature is uniform in space. Spatially varying temperature will lead to energy flows. For small, slow spatial variations the energy current will be proportional to the gradient of the temperature j E = K T with K the thermal conductivity.

17 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, Continuum System: Nonuniform Temperature Macroscopic system is in equilibrium when the temperature is uniform in space. Spatially varying temperature will lead to energy flows. For small, slow spatial variations the energy current will be proportional to the gradient of the temperature j E = K T with K the thermal conductivity. Conservation of energy is given by t ε = j E

18 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, Continuum System: Nonuniform Temperature Macroscopic system is in equilibrium when the temperature is uniform in space. Spatially varying temperature will lead to energy flows. For small, slow spatial variations the energy current will be proportional to the gradient of the temperature j E = K T with K the thermal conductivity. Conservation of energy is given by t ε = j E Law of increase of entropy constrains the coefficient K to be positive.

19 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, Relaxation Mode

20 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, Relaxation Mode Take δε = CδT with C the specific heat per unit volume.

21 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, Relaxation Mode Take δε = CδT with C the specific heat per unit volume. For small temperature perturbations C and K may be taken as constants.

22 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, Relaxation Mode Take δε = CδT with C the specific heat per unit volume. For small temperature perturbations C and K may be taken as constants. The two equations can be combined into the single diffusion equation with diffusion constant κ = K/C. t T = κ 2 T

23 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, Relaxation Mode Take δε = CδT with C the specific heat per unit volume. For small temperature perturbations C and K may be taken as constants. The two equations can be combined into the single diffusion equation with diffusion constant κ = K/C. t T = κ 2 T The results may be extended to the general case of coupled equations for more than one conserved quantity. Sometimes the coupling gives a wave equation rather than diffusion.

24 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, Relaxation Mode Take δε = CδT with C the specific heat per unit volume. For small temperature perturbations C and K may be taken as constants. The two equations can be combined into the single diffusion equation with diffusion constant κ = K/C. t T = κ 2 T The results may be extended to the general case of coupled equations for more than one conserved quantity. Sometimes the coupling gives a wave equation rather than diffusion. Dynamical equations are linear, and the time evolution will be the sum of exponentially oscillating/decaying modes.

25 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, Summary

26 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, Summary A state near equilibrium decays exponentially towards equilibrium, perhaps with transient oscillations.

27 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, Summary A state near equilibrium decays exponentially towards equilibrium, perhaps with transient oscillations. The dynamics is given by equations of motion that are on the one hand the usual phenomenological equations (in our simple example the heat current proportional to temperature difference) and on the other hand consistent with the fundamental laws of thermodynamics.

28 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, Summary A state near equilibrium decays exponentially towards equilibrium, perhaps with transient oscillations. The dynamics is given by equations of motion that are on the one hand the usual phenomenological equations (in our simple example the heat current proportional to temperature difference) and on the other hand consistent with the fundamental laws of thermodynamics. Law of the increase of entropy places constraints on the coefficients of the dynamical equations.

29 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, Summary A state near equilibrium decays exponentially towards equilibrium, perhaps with transient oscillations. The dynamics is given by equations of motion that are on the one hand the usual phenomenological equations (in our simple example the heat current proportional to temperature difference) and on the other hand consistent with the fundamental laws of thermodynamics. Law of the increase of entropy places constraints on the coefficients of the dynamical equations. We are left with the task of calculating kinetic coefficients such as K.

30 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, Onsager s Idea (Regression) Decay to equilibrium from a prepared initial condition is related to dynamics of fluctuations in the equilibrium state equilibrium near equilibrium fluctuations dissipation correlation function kinetic coefficient

31 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, Onsager s Idea (Regression) Decay to equilibrium from a prepared initial condition is related to dynamics of fluctuations in the equilibrium state equilibrium near equilibrium fluctuations dissipation correlation function kinetic coefficient Relationship may be useful in either direction

32 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, Onsager s Idea (Regression) Decay to equilibrium from a prepared initial condition is related to dynamics of fluctuations in the equilibrium state equilibrium near equilibrium fluctuations dissipation correlation function kinetic coefficient Relationship may be useful in either direction Framework of derivation: linear response theory

33 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, Linear Response Theory

34 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, Linear Response Theory Calculate the change in a measurement B(t) due to the application of a small field F(t) giving a perturbation to the Hamiltonian H = F(t)A.

35 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, Linear Response Theory Calculate the change in a measurement B(t) due to the application of a small field F(t) giving a perturbation to the Hamiltonian H = F(t)A. Both A and B are determined by the phase space coordinates r N (t), p N (t). For example, an electric field E = (1/c)dA/dt gives the perturbation H = (e/mc)a(t) p N (t). N

36 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, Linear Response Theory Calculate the change in a measurement B(t) due to the application of a small field F(t) giving a perturbation to the Hamiltonian H = F(t)A. Both A and B are determined by the phase space coordinates r N (t), p N (t). For example, an electric field E = (1/c)dA/dt gives the perturbation H = (e/mc)a(t) p N (t). Time dependence is given by the evolution of r N (t), p N (t) according to Hamilton s equations. N

37 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, Linear Response Theory Calculate the change in a measurement B(t) due to the application of a small field F(t) giving a perturbation to the Hamiltonian H = F(t)A. Both A and B are determined by the phase space coordinates r N (t), p N (t). For example, an electric field E = (1/c)dA/dt gives the perturbation H = (e/mc)a(t) p N (t). Time dependence is given by the evolution of r N (t), p N (t) according to Hamilton s equations. We can calculate averages in terms of an ensemble of systems given by a known distribution ρ(r N, p N ) at t = 0. The expectation value at a later time is then B(t) = dr N dp N ρ(r N, p N )B[r N (t) r N, p N (t) p N ] N

38 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, Linear Response Theory Calculate the change in a measurement B(t) due to the application of a small field F(t) giving a perturbation to the Hamiltonian H = F(t)A. Both A and B are determined by the phase space coordinates r N (t), p N (t). For example, an electric field E = (1/c)dA/dt gives the perturbation H = (e/mc)a(t) p N (t). Time dependence is given by the evolution of r N (t), p N (t) according to Hamilton s equations. We can calculate averages in terms of an ensemble of systems given by a known distribution ρ(r N, p N ) at t = 0. The expectation value at a later time is then B(t) = dr N dp N ρ(r N, p N )B[r N (t) r N, p N (t) p N ] Could alternatively follow the time evolution of ρ through Liouville s equation B(t) = dr N dp N ρ(r N, p N, t)b(r N, p N ) N

39 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, Proof of Onsager Regression: Idea F(t) equilibrium under H 0 + H ρ=ρ H (r N,p N ) H=H 0 + H H=H 0 t We will consider the special case of a force F(t) switched on to the value F 0 in the distant past, and then switched off at t = 0. At t = 0 the distribution is the equilibrium one for the perturbed Hamiltonian We are interested in measurements in the system for t > 0 as it relaxes to equilibrium The dynamics occurs under the unperturbed Hamiltonian

40 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, Onsager Regression: Details

41 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, Onsager Regression: Details For t 0 the distribution is the equilibrium one for the Hamiltonian H(r N, p N ) = H 0 + H ρ(r N, p N ) = e β(h 0+ H) with Tr dr N dp N Tre β(h 0+ H)

42 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, Onsager Regression: Details For t 0 the distribution is the equilibrium one for the Hamiltonian H(r N, p N ) = H 0 + H ρ(r N, p N ) = e β(h 0+ H) with Tr dr N dp N Average of B at t = 0is Tre β(h 0+ H) B(0) = Tre β(h 0+ H) B(r N, p N ) Tre β(h 0+ H)

43 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, Onsager Regression: Details For t 0 the distribution is the equilibrium one for the Hamiltonian H(r N, p N ) = H 0 + H ρ(r N, p N ) = e β(h 0+ H) with Tr dr N dp N Average of B at t = 0is For t 0 the average is Tre β(h 0+ H) B(0) = Tre β(h 0+ H) B(r N, p N ) Tre β(h 0+ H) B(t) = Tre β(h 0+ H) B(r N (t) r N, p N (t) p N ) Tre β(h 0+ H) The integral is over r N, p N,and H = H(r N, p N ). The time evolution is given by H 0.

44 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, Onsager Regression: Details (cont.)

45 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, Onsager Regression: Details (cont.) Expand the exponentials e β(h 0+ H) e β H 0(1 β H) so B(t) = B 0 β[ H B(t) 0 B 0 H 0 ] + O( H) 2 Here 0 denotes the ensemble average for a system with no perturbation, i.e., the distribution ρ 0 = e β H 0/Tre β H 0. In the unperturbed system the Hamiltonian is H 0 for all time, and averages such as B(t) 0 are time independent B 0.

46 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, Onsager Regression: Details (cont.) Expand the exponentials e β(h 0+ H) e β H 0(1 β H) so B(t) = B 0 β[ H B(t) 0 B 0 H 0 ] + O( H) 2 Here 0 denotes the ensemble average for a system with no perturbation, i.e., the distribution ρ 0 = e β H 0/Tre β H 0. In the unperturbed system the Hamiltonian is H 0 for all time, and averages such as B(t) 0 are time independent B 0. Writing A(r N, p N ) = A(0), δ A(t) = A(t) A 0 and use H = F 0 A(0), gives for the change in the measured quantity B(t) =β F 0 δ A(0)δB(t) 0 This result proves the Onsager regression hypothesis.

47 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, Kubo Formula

48 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, Kubo Formula For a general F(t) we write the linear response as B(t) = χ AB (t, t )F(t )dt with χ AB the susceptibility or response function with the properties χ AB (t, t ) = χ AB (t t ) χ AB (t t ) = 0 for t < t χ AB ( f ) = χ AB ( f ) stationarity of unperturbed system causality χ AB(t, t ) real

49 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, Kubo Formula For a general F(t) we write the linear response as B(t) = χ AB (t, t )F(t )dt with χ AB the susceptibility or response function with the properties χ AB (t, t ) = χ AB (t t ) χ AB (t t ) = 0 for t < t χ AB ( f ) = χ AB ( f ) stationarity of unperturbed system causality χ AB(t, t ) real For the step function force turned off at t = 0 B(t) =F 0 0 χ AB (t t )dt = F 0 χ AB (τ)dτ t

50 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, Kubo Formula For a general F(t) we write the linear response as B(t) = χ AB (t, t )F(t )dt with χ AB the susceptibility or response function with the properties χ AB (t, t ) = χ AB (t t ) χ AB (t t ) = 0 for t < t χ AB ( f ) = χ AB ( f ) stationarity of unperturbed system causality χ AB(t, t ) real For the step function force turned off at t = 0 B(t) =F 0 0 χ AB (t t )dt = F 0 χ AB (τ)dτ Differentiating B(t) =β F 0 δ A(0)δB(t) 0 then gives the classical Kubo expression β dt d χ AB (t) = δ A(0)δB(t) 0 t 0 0 t < 0 t

51 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, Energy Absorption

52 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, Energy Absorption Rate of doing work on the system is force velocity W = F Ȧ W = F(t) d dt writing simply χ for χ AA. χ(t, t )F(t )dt

53 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, Energy Absorption Rate of doing work on the system is force velocity W = F Ȧ W = F(t) d dt writing simply χ for χ AA. χ(t, t )F(t )dt For a sinusoidal force F(t) = 1 2 (F f e 2πift + c.c.) the integral gives the Fourier transform χ of χ so that the average rate of working is W ( f ) = 1 4 2πif F f 2 [ χ(f ) χ( f )] = π f F f 2 ( χ ( f )) where χ is Im χ and terms varying as e ±4πift average to zero.

54 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, Energy Absorption Rate of doing work on the system is force velocity W = F Ȧ W = F(t) d dt writing simply χ for χ AA. χ(t, t )F(t )dt For a sinusoidal force F(t) = 1 2 (F f e 2πift + c.c.) the integral gives the Fourier transform χ of χ so that the average rate of working is W ( f ) = 1 4 2πif F f 2 [ χ(f ) χ( f )] = π f F f 2 ( χ ( f )) where χ is Im χ and terms varying as e ±4πift average to zero. The imaginary part of χ tells us about the energy absorption or dissipation.

55 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, Fluctuation-Dissipation Use the fluctuation expression for χ = χ AA χ ( f ) = = β χ(t) sin(2π ft)dt 0 = β(2π f ) d dt δ A(0)δ A(t) 0 sin(2π ft)dt 0 (definition of Fourier transform) (fluctuation expression) δ A(0)δ A(t) 0 cos(2π ft)dt (integrate by parts)

56 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, Fluctuation-Dissipation Use the fluctuation expression for χ = χ AA χ ( f ) = = β χ(t) sin(2π ft)dt 0 = β(2π f ) d dt δ A(0)δ A(t) 0 sin(2π ft)dt 0 (definition of Fourier transform) (fluctuation expression) δ A(0)δ A(t) 0 cos(2π ft)dt (integrate by parts) The integral is the spectral density of A fluctuations, so that (including necessary factors) G A ( f ) = 4k B T ( χ ( f )) 2π f This relates the spectral density of fluctuations to the susceptibility component giving energy absorption.

57 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, Langevin Force

58 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, Langevin Force Suppose the fluctuations in A derive from a fluctuating (Langevin) force F δ A(t) = χ(t, t )F (t )dt

59 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, Langevin Force Suppose the fluctuations in A derive from a fluctuating (Langevin) force F δ A(t) = χ(t, t )F (t )dt Since the Fourier transform of a convolution is just the product of the Fourier transforms the spectral density of A is G A ( f ) = χ(f ) 2 G F ( f )

60 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, Langevin Force Suppose the fluctuations in A derive from a fluctuating (Langevin) force F δ A(t) = χ(t, t )F (t )dt Since the Fourier transform of a convolution is just the product of the Fourier transforms the spectral density of A is G A ( f ) = χ(f ) 2 G F ( f ) Using the expression G A ( f ) = 4k B T ( χ ( f ))/2π f leads to G F ( f ) = 4k B T 1 [ ] 1 2π f Im χ(f )

61 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, Langevin Force Suppose the fluctuations in A derive from a fluctuating (Langevin) force F δ A(t) = χ(t, t )F (t )dt Since the Fourier transform of a convolution is just the product of the Fourier transforms the spectral density of A is G A ( f ) = χ(f ) 2 G F ( f ) Using the expression G A ( f ) = 4k B T ( χ ( f ))/2π f leads to G F ( f ) = 4k B T 1 [ ] 1 2π f Im χ(f ) Instead of the susceptibility introduce the impedance Z = F/Ȧ so that Z( f ) = 1 2πif 1 χ(f )

62 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, Langevin Force Suppose the fluctuations in A derive from a fluctuating (Langevin) force F δ A(t) = χ(t, t )F (t )dt Since the Fourier transform of a convolution is just the product of the Fourier transforms the spectral density of A is G A ( f ) = χ(f ) 2 G F ( f ) Using the expression G A ( f ) = 4k B T ( χ ( f ))/2π f leads to G F ( f ) = 4k B T 1 [ ] 1 2π f Im χ(f ) Instead of the susceptibility introduce the impedance Z = F/Ȧ so that Z( f ) = 1 2πif Defining the resistance R( f ) = Re Z( f ) gives 1 χ(f ) G F ( f ) = 4k B T R( f )

63 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, Quantum Result The derivations have been classical In a quantum treatment A and B,as well as H are operators that may not commute The change to the fluctuation-dissipation is to make the replacement k B T hf hf 2 coth( 2k B T ) so that G F ( f ) = 2hf coth(hf/2k B T ) R( f ) Quantum approach was pioneered by Kubo, and the set of ideas is often called the Kubo formalism.

64 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, BioNEMS - Single BioMolecule Detector/Probe

65 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, BioNEMS Prototype (Arlett et. al, Nobel Symposium 131, August 2005)

66 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, Example Design Parameters b Wall Localized stress L 1 L Localized stress h w y x z Dimensions: L = 3µ, w = 100nm, t = 30nm, L 1 = 0.6µ, b = 33nm Material: ρ = 2230Kg/m 3, E = N/m 2 Results: Spring constant K = 8.7mN/m; vacuum frequency ν 0 6MHz

67 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, Atomic Force Microscopy (AFM)

68 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, Noise in Micro-Cantilevers Thermal fluctuations (Brownian motion) important for: BioNEMS limit to sensitivity detection scheme AFM calibration Goals (with Mark Paul): Correct formulation of fluctuations for analytic calculations Practical scheme for numerical calculations of realistic geometries

69 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, Approach Using Fluctuation-Dissipation Theorem Assume observable is tip displacement X (t) Apply small step force of strength F 0 to tip Calculate or simulate deterministic decay of X (t) for t > 0. Then C XX (t) = δx (t)δx (0) e = k B T X (t) F 0 Fourier transform of C XX (t) gives power spectrum of X fluctuations G X (ω)

70 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, F(t) F 0 H=H 0 + H H=H 0 <X(t)> t δx (t)δx (0) e = k B T X (t) F 0 t

71 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, Advantages of Method Correct! Essentially no approximations in formulation Incorporates full elastic-fluid coupling non-white, spatially dependent noise no assumption on independence of mode fluctuations complex geometries Single numerical calculation over decay time gives complete power spectrum Can be modified for other measurement protocols by appropriate choice of conjugate force AFM: deflection of light (angle near tip) BioNEMS: curvature near pivot (piezoresistivity)

72 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, Single Cantilever b Wall Localized stress L 1 L Localized stress h w y x z Dimensions:L = 3µ, W = 100nm, L 1 = 0.6µ, b = 33nm Material: ρ = 2230Kg/m 3, E = N/m 2

73 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, Device Schematic

74 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, Adjacent Cantilevers Correlation of Brownian fluctuations δx 2 (t)δx 1 (0) e = k B T X 2(t) F 1

75 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, Device Schematic

76 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, Results: Single Cantilever 3d Elastic-fluid code from CFD Research Corporation 1µs force sensitivity: K G X (ν) 1MHz 7pN

77 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, Results: Adjacent Cantilevers Autocorrelation of the noise for Cantilever 1 x 1 (0)x 1 (t) (nm 2 ) x 1 (0)x 2 (t) (nm 2 ) F t(µs) s=5h F 1 s=h Step force initial condition t=0 t Crosscorrelation of the noise for Cantilever t(µs)

78 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, AFM Experiments 1.2 x µ 20.11µ 0.573µ Asylum Research AFM (Clarke et al., 2005) Dashed line: calculations from fluctuation-dissipation approach Dotted line: calculations from Sader (1998) approach

79 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 5, April 14, Next Lecture Hydrodynamics Hydrodynamics of conserved quantities Hydrodynamics of ordered systems (including dissipation)