ExercisesFluctuations in Damped Oscillators.Noise and Langevin equations.2Physics 562: Statistical MechanicsSpring 2005, James P. SethnaHW10 New ProblemsLatest revision: March 24, 2005Exercises(0.1) Fluctuations in Damped Oscillators.Let us explore further the fluctuating mass-on-a-springexample of section 6.5. We saw there that the couplingof the macroscopic motion to the internal degrees of free-dom will eventually damp the oscillations: the remain-ing motions were microscopic for macroscopic springs andmasses. These remanent thermal fluctuations can be im-portant, however, for nanomechanical and biological sys-tems. In addition, the damped harmonic oscillator is aclassic model for many other physical processes, such asdielectric loss in insulators. (See [72] for a nice treatmentby an originator of this subject.)Consider a damped, simple harmonic oscillator, forcedwith an external force f, obeying the equation of motiond2θdt2= −ω20θ2− γdθdt+ f(t)/m. (1)(a) Susceptibility. Find the AC susceptibility eχ(ω) forthe oscillator. Plot χ0and χ00for ω0= m = 1 andγ = 0.2, 2, and 5. (Hint: Fourier transform the equationof motion, and solve foreθ in terms ofef.)(b) Causality and Critical Damping. Check, for pos-itive damping γ, that your χ(ω) is causal (χ(t) = 0 fort < 0), by examining the singularities in the complex ωplane (section 10.6). At what value of γ do the poles beginto sit on the imaginary axis? The system is overdamped,and the oscillations disappear, when the poles are on theimaginary axis.At this point, it would be natural to ask you to verifythe Kramers-Kr¨onig relation, equation 10.72, and showexplicitly that you can write χ0in terms of χ00. Thatturns out to be tricky both analytically and numerically,though: if you’re ambitious, try it.(c) Dissipation and Susceptibility. Given a forcingf(t) = A cos(ωt), solve the equation and calculate θ(t).Calculate the average power dissipated by integrating yourresulting formula for f dθ/dt. Do your answers for thepower and χ00agree with the general formula for powerdissipation, equation 10.35?(d) Correlations and Thermal Equilibrium. Usethe fluctuation–dissipation theorem to calculate the cor-relation function˜C(ω) from χ00(ω), (see equation 10.64),whereC(t − t0) = hθ(t)θ(t0)i. (2)Find the equal-time correlation function C(0) = hθ2i, andshow that it satisfies the equipartition theorem. (Hints:our oscillator is in a potential well V (θ) =1/2mω20θ2. Youwill need to know the integralR∞−∞1ω2+(1−ω2)2dω = π.)(0.2) Noise and Langevin equations.In this chapter, we have never explicitly discussed how theenergy removed from a system by damping is returned tothe system to maintain thermal equilibrium. This energyinput is through the thermal fluctuation noise introducedthrough the coupling to the heat bath. In this exercise wewill derive a Langevin equation incorporating both noiseand dissipation (see also [20, section 8.8]).We start with a system with coordinate P, Q and internalpotential energy V (Q), coupled linearly to a heat baththrough some coupling term Q · F:H =P22m+ V (Q) + Hbath(y1, y2, y3, . . . ) − Q · F(y1, . . . ).(3)In the absence of the coupling to our system, assume thatthe bath would contribute an external noise Fb(t) withmean zero. In the presence of the coupling to the sys-tem, the mean value of the force will develop a non-zeroexpectation valuehF(t)i =Zt−∞dt0χb(t − t0)Q(t0), (4)where χb(t−t0) is the susceptibility of the bath to the mo-tion of the system Q(t). Our system then has an equationof motion with a random noise F and a time-retarded in-teraction due to χb:m¨Q = −∂V∂Q+ Fb+Zt−∞dt0χb(t − t0)Q(t0). (5)To be pub. Oxford UP, ∼Fall’05 www.physics.cornell.edu/sethna/StatMech/3We can write this susceptibility in terms correlation func-tion of the noise in the absence of the systemCb(t − t0) = hFb(t)Fb(t0)i (6)using the fluctuation-dissipation theoremχb(t − t0) = −β∂Cb∂t. t > t0. (7)(a) Integrating by parts and keeping the boundary terms,show that the equation of motion has the formm¨Q = −∂¯V∂Q+ Fb−Zt−∞dt0Cb(t − t0)˙Q(t0). (8)What is the ‘potential of mean force’¯V , in terms of Vand Cb?(b) If the correlations in the bath are short-lived com-pared to the time-scales of the system, we can approximate˙Q(t0) ≈˙Q(t) in equation 8, leading to a viscous frictionforce −γ˙Q. What is the formula for γ?(c) Conversely, for a model system with a perfect viscousfriction law −γ˙Q at temperature T , derive the equationfor correlation function for the noise Cb(t − t0). No-tice that viscous friction implies a memoryless, Markovianheat bath, and vice-versa.Langevin equations are useful both in analytic calcula-tions, and as one method for maintaining a constant tem-perature in molecular dynamics simulations.cJames P. Sethna, March 24, 2005 Entropy, Order Parameters, and ComplexityBibliography[1] Guenter Ahlers. Critical phenomena at low temperatures. Rev.Mod. Phys., 52:489–503, 1980.[2] Vinay Ambegaokar, B. I. Halperin, David Nelson, and Eric Sig-gia. Dissipation in two–dimensional superfluids. Physical ReviewLetters, 40:783–6, 1978.[3] Vinay Ambegaokar, B. I. Halperin, David Nelson, and Eric Siggia.Dynamics of superfluid films. Physical Review B, 21:1806–26, 1980.[4] M. H. Anderson, J.R. Ensher, M.R. Matthews, C.E. Wieman, andE.A. Cornell. Observation of bose-einstein condensation in a diluteatomic vapor. Science, 269, 1995. jilawww.colorado.edu/bec/ .[5] P. W. Anderson. Coherent matter field phenomena in superfluids.In Some Recent Definitions in the Basic Sciences, volume 2. BelferGraduate School of Science, Yeshiva University, New York, 1965.Reprinted in [7, p. 229].[6] P. W. Anderson. Considerations on the flow of superfluid helium.Rev. Mod. Phys., 38, 1966. Reprinted in [7, p. 249].[7] P. W. Anderson. Basic Notions of Condensed Matter Physics.Benjamin–Cummings, Menlo Park, CA, 1984.[8] Neil W. Ashcroft and N. David Mermin. Solid State Physics. Hold,Rinehart, and Wilson, Philadelphia, 1976.[9] Carl M. Bender and Steven A. Orszag. Advanced MathematicalMethods for Scientists and Engineers. McGraw–Hill, New York,1978.[10] Howard Berg. Motile behavior of bacteria. Physics Today, 53:24–29, 2000. www.aip.org/pt/jan00/berg.htm .[11] N. O. Birge and S. R. Nagel. Specific heat spectroscopy of theglass transition. Physical Review Letters, 54:2674, 1985.[12] Eberhard Bodenschatz, Brian Utter, and