Physics 17b: Statistical Mechanics Linear Response Theory Useful references are Callen and Greene [1], and Chandler [], chapter 16. Task To calculate the change in a measurement B t) due to the application of a small field F t) that gives a perturbation to the Hamiltonian H = F t) A. Here both A and B are determined by the phase space coordinates denoted r N t), p N t) e.g. B might be the electric current N i=1 e/m) p i). The time dependence is given by the evolution of r N t), p N t) according to Hamilton s equations. In statistical mechanics we deal with an ensemble of systems given for example by a known distribution ρ r N, p N) at t = 0. The expectation value at a later time t is then B t) = d r N d p N ρ r N, p N) B r N t) r N, p N t) p N) 1) where the cumbersome notation r N t) r N means we must evaluate B at the phase space coordinate that evolves from the value r N at t = 0. For shorthand this will be denoted B t). We could equivalently follow the time evolution of ρ through Liouville s equation and instead evaluate B t) = d r N d p N ρ r N, p N,t ) B r N, p N) ) but the first form is more convenient. Onsager regression Ft) equilibrium under H 0 + H ρ=ρ H r N,p N ) H=H 0 + H H=H 0 t First consider the special case of a force F t) switched on to the value F in the distant past, and then switched off at t = 0. We are interested in measurements in the system for t>0 as it relaxes to equilibrium. For t 0 the distribution is the equilibrium one for the perturbed Hamiltonian H r N, p N) = H 0 + H, with H 0 the unperturbed Hamiltonian, i.e. for a canonical distribution) ρ r N, p N) = e βh 0+ H ) d rn d p N e βh 0+ H ) 3) 1
so that introducing a convenient notation for the integral over phase space Tr d r N d p N B 0) = Tre βh 0+ H ) B r N, p N) Tre βh 0+ H ). 4) For t 0 we let r N t), p N t) for each member of the ensemble evolve under the Hamiltonian, now H 0 from its value r N, p N at t = 0,so that B t) = Tre βh 0+ H ) B r N t) r N, p N t) p N). 5) Tre βh 0+ H ) Note that the integral is over r N, p N which we can denote r N 0), p N 0), and H = H r N, p N) etc. It is now a simple matter to expand the exponentials to first order in H F small!) B t) Tre βh 0 1 β H)B r N t) r N, p N t) p N) 6) Tre βh 0 1 β H) to give B t) = B 0 β [ H B t) 0 B 0 H 0 ] + O H ) where <> 0 denotes the average over the ensemble for a system for which no perturbation was applied i.e. ρ 0 = e βh 0 /Tre βh 0. Note in this unperturbed system the Hamiltonian remains H 0 for all time, so that one-time averages such as B t) 0 are in fact time independent. Finally putting in the form of H, writing δb t) = B t) B 0 etc., and noticing that A r N, p N ) is equivalent to A0), H = FA r N, p N) = FA r N 0), p N 0) ) = FA0), 8) 7) gives for the change in the measurement B t) = B t) B 0 B t) = βf δa 0) δb t) 0, 9) where we have written the result in terms of δa = A A 0. This result which equates the time dependence of the decay of a prepared perturbation to the time dependence of a correlation function in the unperturbed system proves the Onsager regression hypothesis. Kubo formula For a general F t) we write the linear response as B t) = χ AB t,t ) F t ) dt 10) with χ AB the susceptibility or response function with the properties χ ) AB t,t = χ ) AB t t stationarity of unperturbed system χ ) AB t t = 0 for t<t causality χ AB f ) = χ AB f ) ) t,t real χ AB. 11) The causality condition leads to interesting properties in the complex f plane: for the sign convention I am using for the Fourier transforms, χ f ) must be analytic, has no poles) in the upper half plane. We will not need this result here.
For the step function force turned off at t = 0 B t) = F = F 0 Differentiating then gives the classical Kubo expression t χ ) AB t t dt 1a) χ AB τ) dτ. 1b) χ AB t) = { β d dt δa 0) δb t) 0 t 0 0 t<0. 13) Connection with energy absorption: fluctuation-dissipation Consider a sinusoidal force F t) = Re F f e πift = 1 F f e πift + c.c.). The rate of doing work on the system is force velocity W = F Ȧ i.e. W = F t) d dt χ t,t ) F t ) dt 14) writing simply χ for χ AA. Substituting in the sinusoidal form for the force, and remembering that χt,t ) only depends on the time difference τ = t t W = 1 d ) 4 F f e πift + c.c.) dt e πift χ τ) F f e πifτ dτ + c.c. 15) we recognize the integral as giving the Fourier transform of χ, so that the rate of working averaged over a period is W = 1 4 πif Ff [ ] χ f ) χ f ) 16a) = πf Ff χ f ) 16b) where it is conventional to write χ = χ + i χ with χ the real part Re χ not the driviative!) and χ the imaginary part Im χ, and terms varying as e ±4πift average to zero. Thus the imaginary part of χ tells us about the energy absorption or dissipation. But from the definition of the Fourier transform χ f ) = = β χ t) sin πf t) dt 0 = β πf ) d dt δa 0) δa t) 0 sin πf t) dt 0 δa 0) δa t) 0 cos πf t) dt 17a) 17b) where we have used the Kubo expression Eq. 13) for χ = χ AA and then integrated by parts. Finally we recognize the integral as giving the spectral density of A fluctuations, so that including necessary factors) 17c) G A f ) = 4k B T χ f ) πf 18) relating the spectral density of fluctuations to the susceptibility component giving energy absorption, i.e. the classical fluctuation-dissipation theorem. 3
Langevin Force If we suppose the fluctuations in A derive from a fluctuating Langevin contribution F to the force F we can write A t) = χ t,t ) F t ) dt. 19) Since the Fourier transform of a convolution is just the product of the Fourier transforms the spectral density of At) is given by G A f ) = χ f ) G F f ), 0) and using the expression for G A derived above leads to G F f ) = 4k B T 1 πf Im [ 1 ] χ f ) Instead of the susceptibility we can introduce the impedance Z = F/ Ȧ so that. 1) Z f ) = 1 πif 1 χ f ). ) Then defining the resistance R f ) = Re Z f ) we can write the spectral density of the fluctuating force as G F f ) = 4k B T R f ). 3) The analogy with the Johnson noise expression should be apparent. Note that the derivations have been classical, so we arrive at the classical version of the fluctuationdissipation theorem, the force expression etc. In a quantum treatment A and B, as well as H are operators that may not commute, so that we have to be more careful in the expansion of the exponentials. Thus Eq. 13) will involve commutators of operators. The simplest form of the expression is to define a slightly different susceptibility χ t) = i) 1 [χt) χ t)] the Fourier inverse of the dissipative part χ f )) which is then related to the commutator-correlation function χ AB t) = 1 h [A0), Bt)] 0, 4) where A, B are Heisenberg time dependent) operators. The change to Eq. 18) is quite simple: make the replacement k B T hf ) hf coth 5) k B T which can be interpreted as taking into account the Bose occupation factor of the modes. The quantum approach was pioneered by Kubo [3], and the set of ideas are often called the Kubo formalism. Applications to LIGO The fluctuation-dissipation theorem is an important tool in understanding the noise that limits the sensitivity of the LIGO experiment at the operation frequency thermal noise is the dominant source. Therefore: understand the dissipation and you understand the noise. Some references that should now be readable are: 1. A. Abramovici et al., Science 56, 35 199) general review of LIGO. P.R. Slauson, Phys. Rev. D4, 437 1990) discussion of mechanical dissipation and noise available online) 4
3. A. Gillespie and F. Raab, Phys. Lett. A190, 13 1994) theory and measurements of suspension losses and noise in a LIGO test bed 4. Y. Levin, Phys. Rev. D57, 659 1998) application of the fluctuation dissipation theorem leading to a novel calculation of mass-mode noise available online) The paper by Yuri Levin, who was a student here a few years ago, is a particularly nice application of the ideas we have been talking about, and I strongly urge you to read it. References [1] H.B. Callen and R.F. Greene, Phys. Rev. 86, 70 195) [] D. Chandler, Introduction to modern statistical mechanics Oxford University Press, 1987) [3] R. Kubo, J. Phys. Soc. Japan 1, 570 1957) 5