Duality and Reciprocity of Fluctuation-Dissipation Relations In

arXiv:1606.01659v1 [cond-mat.stat-mech] 6 Jun 2016 h aineo urn rvlaeflcutos measured fluctuations) voltage or (i.e., current response of the macroscopic variance and the the of system), variance physical the the is as response in- (taken fluctuations conductor spontaneous a perturbation of side source the microscopic Here the fluc- is equilibrium. electrical thermal for at relation tuations reciprocity and property dual [1]. analyze to complex simpler a something reducing to of down methods linear-network electrotech- valid equally are in two properties are that nics above theorems the Norton’s of Thévenin’s applications and used most The ihtercpoiyrlto ie by given relation reciprocity the with (conductance coefficients resistance kinetic interchanged or associated be can the drop) with voltage or current (measured ntelna-epnergm set htperturbation that asserts voltage regime (applied linear-response the in eae.Acrigt h’ a,frahomogeneous a for gives law, property Ohm’s dual the to conductor According related. h i fti etri ofruaea analogous an formulate to is letter this of aim The Introduction iatmnod nenradl’Invzoe Universit` Innovazione, dell’ Ingegneria di Dipartimento ult n eirct fflcuto-ispto relati fluctuation-dissipation of reciprocity and Duality ASnmes 54.a 54.a 72.70.+m 05.40.Ca; va 05.40.-a: which numbers: noise, PACS current for not but vel diffusion single-carrier pro for non-vanishing is microsco responsible it Fermions the particular for on In light principle, conductors. new in sheds phenomena statistics developed fluctuation degenerate so for as theory well unifying as classical no for voltage formulation valid (for The are canonical and relation. noise) current reciprocity (for a canonical mi and the property p Analogously, dual dual resistance. a a noise plasma display corresponding reciprocal to a found the admits are to fluctuations way of voltage res exclusive and the sources current mutual and microscopic a fluctuations voltage in the and related as current the conductor of a frequency to inside individuate inc carriers we relations purpose of this Nyquist For by fluctuations. expressed respective fluctuations voltage and R yaaoywt ierrsos efruaetedaiyan duality the formulate we linear-response with analogy By h ulpoet neetia transport electrical in property dual The - epciey,bigrcpoal inter- reciprocally being respectively), , I V = ripsdcurrent imposed or GV GR and 1 = nvri` e aet,vaMneoi -30 ec,Ita Lecce, I-73100 Monteroni, via Salento, Università del NS,VadlaVsaNvl,8 04 oa Italy Roma, 00146 - 84 Navale, Vasca della Via CNISM, NS,VadlaVsaNvl,8 04 oa Italy Roma, 00146 - 84 Navale, Vasca della Via CNISM, iatmnod aeaiaeFsc,“ni eGiorgi”, de “Ennio Fisica, e Matematica di Dipartimento V . ntttfür Festkörpertheorie, Münster Universität Institut ihl-lm-t.1,419Muse,Germany Münster, 48149 10, Wilhelm-Klemm-Str. = RI, I n response and ) Dtd coe ,2018) 9, October (Dated: looaAlfinito Eleonora ioReggiani Lino imn Kuhn Tilmann (2) (1) G e aet,vaMneoi -30 ec,Iayand Italy Lecce, I-73100 Monteroni, via Salento, del a e ntesml sfie,tu eern otecanonical the to num- referring thus carrier fixed, the is when sample circuit the in open mea- the ber outside is to the noise referring in Voltage thus sured (GCE). sample, ensemble the canonical leave grand carri- or outside where enter the system may in open ers measured an statisti- implies is which appropriate noise short-circuit, Current the of ensemble. modeling, choice cal microscopic the in the determine reflected for which is conditions or circuit current boundary outside of the the measurement in the fluctuations hand, model voltage other the microscopic cir- On equivalent the well-defined cuit. a hand, implies one transport carrier the for prime On of becomes definition importance. system correct a level kinetic of laws addressed basic the be of mechanics. will framework statistical the issues in these solved formally all to and Here, available not knowledge. is our functions, and distribution Fermi-Dirac Bose-Einstein explicitly de- including and a classical statistics, Ultimately, to generate generally [2–6]. applies literature that theory basic unifying controversial, the sometimes in simple even and the physics answers partial, statistical beyond only noise in received thus issue the major that and of a is level, individuation model, kinetic temperature The a at circuit. sources outside the in o h nlsso urn rvlaeflcutoso a on fluctuations voltage or current of analysis the For nldn rcinlecuinsaitc.The statistics. exclusion fractional including ct utain tzr eprtr are temperature zero at fluctuations ocity s)esmlsadrslsaedrvdwhich derived are results and ensembles ise) ∗ a-ubradditvlct fluctuations drift-velocity and tal-number rsoi os-ore r on oobey to found are noise-sources croscopic ihsi hslimit. this in nishes e ht sacneuneo h Pauli the of consequence a as that, ven scridoti h rm ftegrand the of frame the in out carried is uigteitiscbnwdh fthe of bandwidths intrinsic the luding etv odcac n eitneare resistance and conductance pective oet i lsacnutnethat conductance plasma a via roperty suc.Temcocpcvrac of variance macroscopic The -source. os.Teseta este tlow at densities spectral The noise. i nepeaino ispto and dissipation of interpretation pic eirct rpriso current of properties reciprocity d , yand ly n nconductors in ons 2 ensemble (CE). While it is well-known that in the thermodynamic limit different statistical ensembles become equivalent [7], this does not hold anymore in the case of fluctuations, when a finite system size has to be considered. Nevertheless we will show that the dual property provides a direct link between the noise sources in the GCE and the CE. Theory - Electrical fluctuations of a conductor in the limit of low frequency (i.e., ω 0) are described by the Nyquist relations [3] → FIG. 1. Equivalent circuit with impedance Z(ω) of a real I2 homogenous conductor of resistance R. The capacitance C SI (ω =0)=4 =4KBTG, (3) L ∆fI and inductance account for the presence of the contacts and for the inertia of carriers, respectively. V 2 SV (ω =0)=4 =4KBTR, (4) ∆fV we assume a homogeneous conductor with length L in where S and S are the spectral densities of instan- I V x-direction and cross-section A. The link between micro- taneous current and voltage fluctuations, respectively, scopic and macroscopic picture is provided by the Ramo- I2 and V 2 the variances of the corresponding current Shockley theorem [8, 9], which for our geometry reads and voltage fluctuations, (we recall that being at thermal equilibrium their average values are identically zero), d L N(t)q ∆fI and ∆fV the corresponding intrinsic bandwidths de- V (t)= vd(t) I(t) , (5) dt ε0εrA L − termined by the decay of the corresponding correlation functions, KB is Boltzmann’s constant and T the ab- where ǫ0 and ǫr are the vacuum and the relative dielectric solute temperature. Here and henceforth, the bar over constant of the host lattice material, respectively, and physical quantities denotes the ensemble average. The 1 1 dual property we are interested in refers to the above v (t)= v (t)= vk nk(t) (6) d N(t) i,x N(t) ,x Nyquist relations, also called fluctuation-dissipation the- i k orems (FDTs) [5]. X X To describe the real conductor on a macroscopic level is the instantaneous drift velocity in x-direction of the the ideal resistance R has to be complemented by a ki- N(t) carriers with charge q and effective mass m moving v netic inductance associated with the inertia of the car- with velocities i(t) in the conductor, the second form be- riers and a capacitanceL associated with the contacts ing written in terms of the fluctuating occupation number k resulting in the equivalentC circuit shown in Fig. 1. Note of carriers nk(t) in the state with the velocity compo- that current and voltage fluctuations are measured under nent vk,x in this state, which explicitly accounts for the different operation conditions: Voltage noise is measured indistinguishability of the carriers. Extensions of the the- at the contacts in the open circuit (i.e., for I(t) = 0) while orem to more general boundary conditions and quantum current noise is measured in the outside short circuit (i.e., mechanical currents can be found in Refs. [10, 11]. We for V (t) = 0). notice that under voltage noise operation (CE) N(t)= N While the open circuit, being characterized by a closed with N the fixed number of carriers inside the conductor system with no particle exchange, is a well-defined con- and, for average quantities, the usual assumption N = N cept, the short circuit deserves some more comments. In is well justified [7]. This microscopic model indeed leads order to be an ideal short circuit, the resistance of the to the equivalent circuit shown in Fig. 1 and, using kinetic external circuit has to be negligible compared to the re- theory and a Drude model for the carriers, its lumped ele- sistance of the conductor under consideration. The resis- ments are related to the microscopic properties according tance of a material is inversely proportional to the mo- to mentum relaxation time and the number of carriers [see, L2m L2m A e.g., Eq. (7)]. Since the momentum relaxation time typi- R = , = , = ǫ0ǫr , (7) q2Nτ L q2N C L cally cannot be varied to a large extent (unless extremely low temperatures are considered), the ideal short circuit with τ being the momentum relaxation time. Note that requires a very large number of carriers, such that it in- on the microscopic level the different operation condi- deed can be treated as a reservoir in the sense of the tions are reflected in different boundary conditions for GCE. the carriers. Voltage noise operation refers to a closed On a microscopic level current and voltage fluctuations system with fixed carrier number, while current noise op- are generated by the stochastic motion of the carriers in eration refers to an open system, where carriers enter or the conductor. For the reason of clarity of the model leave the system through the contacts. 3

2 Within a correlation function scheme the intrinsic where δvd is the variance of the fluctuations of the in- bandwidths are directly related to the decay of the corre- stantaneous carrier drift-velocity averaged over the sam- lation functions associated with the current and voltage ple, vk,x =¯hkx/m is the x-component of its velocity, εk 2 2 2 fluctuations. In the case of current noise, ∆fI = 1/τ = is the corresponding energy, δf (εk) = nk nk is the − R/ is determined by the momentum relaxation time τ variance of the occupation number and f(εk) = nk is L or, equivalently, by the R time constant of the equiva- the equilibrium distribution-function normalized to car- L lent circuit. Analogously, the intrinsic bandwidth of volt- rier number which, according to statistics, satisfies the age fluctuations ∆fV =1/τd =1/(R ) is determined by property 2C the dielectric relaxation-time τd = τp /τ or, equivalently, k by the R time constant of the circuit. Here, τp denotes 2 ∂f(ε ) C δf (εk)= KBT , (12) the plasma time (i.e., the inverse of the plasma frequency − ∂εk

ωp) 2 Using the symmetry of the problem, vk,x can be replaced by (2εk/md), where d denotes the dimension of the sys- ǫ ǫ mAL −1 0 r √ tem. With the density-of-states of a d-dimensional carrier τp = ωp = 2 = (8) s Nq LC gas satisfying (ε) ε(d/2)−1, Eq. (11) can be evaluated to D ∝ being related to the time constant of the circuit. We LC K T remark that also outside the static limit (i.e., for ω = 0) δv2 = B , (13) the equivalent circuit in Fig. 1 gives the impedance6 (or d m N the admittance) whose real parts reproduce the frequency independent of the dimension d. We remark that since dependence of the current and voltage spectra in the clas- these noise sources are directly obtained from the prop- sical limit K T ¯hω, with ¯h being the reduced Planck B erties of the statistical ensembles, they hold for carriers constant. These≫ spectra are associated with the micro- obeying any type of statistics, in particular for Fermi- scopic time scales of the corresponding correlation func- Dirac (FD) and for Bose-Einstein (BE) statistics, but tions, as was validated by Monte Carlo simulations [12]. also for carriers obeying a fractional exclusion statistics Furthermore, the equivalent circuit consistently recovers (FES) [14, 15]. In all cases Maxwell-Boltzmann (MB) the standard relations statistics is implicitly recovered in the limit f 1. By using Eq. (10) to replace the temperature≪ in Eq. (3) 2 KBT 2 KBT V = and I = . (9) the current fluctuations can be expressed as C L which in this form are valid for any type of inductance G ∂µ0 I2 = δN 2 . (14) and capacitance in a circuit as given in Fig. 1. As such, τ ∂N this circuit is of most physical importance and should replace alternative equivalent circuits (like, e.g., simple R By taking G from the generalized Einstein relation [16] C parallel and R serial circuits) which are also sometimes q 2 ∂N used in the literature.LC G = D , (15) L ∂µ0 Making use of statistics, the temperature can be associated with the microscopic quantities defining two noise where sources that are present in the conductor, one for each 2′ operation condition. These sources should be taken as D = vx τ (16) mutually exclusive for each of the two boundary conditions. Accordingly, constant-voltage (i.e., current noise) is the longitudinal-diffusion coefficient [16] with the dif- operation is associated with a GCE reflecting the open ferential (with respect to carrier number) quadratic ve- system where [13] locity component along the x-direction given by k 2′ 2 ∂f(ε ) v = vk 2 ∂N x ,x δN = KBT . (10) k ∂N ∂µ0 X 2 ∂f(εk) ∂µ0 KBT N 2 = vk = , (17) Here, δN is the variance of the instantaneous number of ,x 2 k ∂µ0 ∂N m δN carriers inside the sample, and µ0 the chemical potential. X Constant-current (i.e., voltage noise) operation, on the where the last equality, obtained in the same way as other hand, is associated with a CE with fixed carrier Eq. (13), follows from Eq. (11) and additionally using number (N = N) and Eq. (10). Notice the explicit appearance of the Fano factor, δN 2/N, to account for the effective interaction 1 2 2 2 k among carriers due to the symmetry properties of their δvd = 2 vk,x δf (ε ), (11) N k wave functions and thus for the correct statistics. X 4

Then, Eq. (14) takes the equivalent forms The dual property of the macroscopic FDTs is obtained from Eqs. (18), (19) and (24) as 2 2′ 2 2 2 2 KBT q vx δN q δN I = = 2 = 2 (18) 2 2 2 2 2 2 L τ I = Gp V and V = Rp I (25) L N 2 2 2′ with a plasma conductance Gp = (qNµp)/L and a with τN = L / vx being an effective transport-time plasma resistance Rp satisfying the reciprocity relation through theq sample [17] determining the conversion of total-number fluctuations of carriers inside the sample GpRp =1 . (26) into total-current fluctuations measured in the external short-circuit. By satisfying the relations (3) and (4), the expressions For the variance of voltage fluctuations, substitution (25) and (26) justify the identification of the intrinsic of Eq. (13) into Eq. (4) gives the equivalent expressions bandwidths here assumed. Equations (25) and (26) express the duality and reci- 2 2 2 2 2 KBT mNLδvd m L δvd procity properties of fluctuation-dissipation relations and V = = = 2 2 (19) Aǫ0ǫr q τ parallels Eqs. (1) and (2) of linear-response relations. No- C p tice, that all the above expressions hold for any type of with the plasma time τp given in Eq. (8). Introducing statistics (in the case of Bosons for temperatures above the variance of electric-field fluctuations averaged over the critical temperature for Bose-Einstein condensation 2 2 2 the sample length δE = δV /L and the plasma carrier- [18]), thus complementing the standard FDT in the limit mobility µp = qτp/m, Eq. (19) can be written in terms of low frequencies. From statistics, the two boundary of a generalized Ohm’s law conditions are associated with a GCE and a CE, respec- 2 2 2 tively, and Eq. (24) shows the interesting results that δvd = µp δE , (20) both statistics provide the same result even outside the describing the conversion of carrier drift-velocity fluctu- thermodynamic limit conditions [7]. ations inside the sample into electric field (or voltage) Conclusions - We have formulated the dual property fluctuations at the terminals of the open circuit. and the reciprocity relation of the FDTs in the limit of By using Eqs. (3) and (10) the macroscopic conduc- low frequency by expressing conductance and resistance tance is associated with carrier total-number fluctuations in terms of the proper microscopic noise-source, what we by: call the dissipation-fluctuation relations. Analogously, by accounting for the intrinsic bandwidth, the variances of q2v2′ τ x 2 current and voltage fluctuations are related to the same G = 2 δN . (21) L KBT noise-sources, what are usually called the fluctuation dis- Analogously, using Eqs. (4) and (13) the macroscopic re- sipation relations. All these relations are given in a form sistance is associated with drift-velocity fluctuations by: that is independent of the type of distribution functions, thus including MB, FD and BE statistics (the latter at 2 2 L m 2 temperatures above Bose-Einstein condensation) as well R = 2 δvd. (22) q τKBT as FES. From a physical point of view, the temperature entering the Nyquist relations (3) and (4) is here From a microscopic point of view, Eqs. (2), (21) and (22) expressed in kinetic terms and associated with the vari- imply that the noise sources satisfy the relations ance of instantaneous fluctuations (i) of the total number 2′ 2 2 2 2′ 2 of carriers inside the sample, or (ii) of the carrier drift- vx δN m δvd vx δN GR = 2 = 2 = 1 (23) velocity inside the sample. Calculations are carried out (KBT ) δv2 N d in the framework of the GCE and the CE and the result and thus summarized in Eq. (24) provides an interesting example where both ensembles give the same result even outside 2 2′ 2 2 NKBT δN vx = N δvd = , (24) the thermodynamic limit. m While thermal noise and shot noise are typically de- where again Eq. (13) has been used to eliminate the tem- scribed as different noise phenomena, one originating perature in Eq. (23). from the thermal agitation of the carriers [2, 3] and the Equations (23) and (24) express the reciprocity and other from the discreteness of the charge [19–21], our re- duality properties of the microscopic noise-sources as- sults clearly indicate the close relationship between both sociated with the fluctuation-dissipation relations. In of them. Indeed, by expressing current fluctuations in other words, at thermodynamic equilibrium carrier total- terms of the ratio between the variance of carrier num- number fluctuations inside a conductor under constant- ber fluctuations and the effective transport time τN we voltage conditions are inter-related to carrier drift- show that the source of shot-noise is already present at velocity fluctuations under constant-current conditions. thermal equilibrium [22]. 5

The present formulation shows that for Fermions the [6] Yu. L. Klimontovich, Fluctuation-dissipation relations. vanishing of the low-frequency current spectral-density Role of the finiteness of the correlation time. Quantum at T = 0 is associated with the vanishing of the vari- generalization of Nyquist’s formula, Sov. Phys. Usp. 30, ance of carrier total-number fluctuations, i.e., with the No. 2, 154 (1987). [7] P. T. Landsberg, Quantum Statistics of Closed and Open instantaneous correlation between appearance and dis- Systems, Phys. Rev. 93, 1170 (1954). appearance of an elemental carrier number fluctuation [8] W. Shockley, Currents to Conductors Induced by a Mov- inside the sample. This is essential because for Fermions ing Point Charge, J. Appl. Phys. 9, 635 (1938). the value of the diffusion coefficient [Eq. (16)] does not [9] S. Ramo, Currents Induced by Electron Motion, Proceed- vanish at T = 0, and thus the notion of diffusion as syn- ings of the IRE 27, 584 (1939). onymous of noise, fails completely. In contrast, for the [10] B. Pellegrini, Electric charge motion, induced current, 34 classical case the absence of motion at T = 0, i.e., D =0 energy balance, and noise, Phys. Rev. B , 5921 1986). [11] B. Pellegrini, Extension of the Electrokinematics The- and τ , is definitely responsible for the vanishing N → ∞ orem to the Electromagnetic Field and Quantum Me- of current noise. chanics, Nuovo Cimento D 15, 855 (1993); ibid. Elemen- For the case of voltage fluctuations, the vanishing of tary Applications of Quantum-Electrokinematics Theo- thermal noise at T = 0 is associated with the tendency rem, 881 (1993). of the drift-velocity fluctuations to approach zero, which [12] L. Reggiani, All the colors of noise, in 22nd International is a property independent of the kind of statistics. Conference on Noise and Fluctuations (ICNF), IEEE As a final remark, we notice that the equivalent circuit 10.1109/ICNF.2013.6578874, 1 (2013). here introduced reproduces the correct frequency depen- [13] M. Lax, Fluctuations from the Nonequilibrium Steady State, Rev. Mod. Phys. 32, 25 (1960). dence of both current and voltage spectral densities, the [14] B. I. Halperin, Statistics of Quasiparticles and latter one including the plasmonic contribution in the the Hierarchy of Fractional Quantized Hall States, case τd τ [12]. We want to stress that the time scales Phys. Rev. Lett. 52, 1583 (1984). ≪ of the fluctuating macroscopic variables are in general not [15] Y.-S. Wu, Statistical Distribution for General- related to those of the respective noise sources. Instead, ized Ideal Gas of Fractional-Statistics Particles, they should be treated in the framework of the time or Phys. Rev. Lett. 73, 922 (1995). frequency dependence of the corresponding correlation [16] S. Gantsevich, R. Katilius and V. Gurevich, Theory of fluctuations in nonequilibrium electron gas, Rivista functions or spectral densities. Nuovo Cimento 2, 1 (1979). [17] A. Greiner, L. Reggiani, T. Kuhn and L. Varani, Car- rier kinetics from the diffusive to the ballistic regime: linear response near thermodynamic equilibrium, Semi- 15 ∗ cond. Sci. Technol. , 1071 (2000). [email protected] [18] J. Dunning-Davis, Particle-number fluctuations, Nuovo [1] D. H. Johnson, Origin of the Equivalent Circuit: The Cimento 57B, 315 (1968). 91 Voltage-Source Equivalent, Proc. IEEE , 636 (2003); [19] W. Schottky, Uber¨ spontane Stromschwankungen in ver- 91 The Current-Source Equivalent, Proc. IEEE , 817 schiednenen Elektrizitätsleitern, Ann. Phys. 362, 541 (2003). (1918). [2] J. B. Johnson, Thermal Agitation of Electricity in Con- [20] Ya. M. Blanter and M. Büttiker, Shot noise in mesoscopic 32 ductors, Phys. Rev. , 97 (1928). conductors, Physics Reports 336, 1 (2000). [3] H. Nyquist, Thermal Agitation of Electric Charge in Con- [21] C. Beenakker and C. Schönenberger, Quantum shot 32 ductors, Phys. Rev. , 110 (1928). noise, Physics Today 56, No. 5, 33 (2003). [4] H. B. Callen and T. A. Welton, Irreversibility and Gen- [22] G. Gomila, C. Pennetta, L. Reggiani, M. Sampietro, 83 eralized Noise, Phys. Rev. , 34 (1951). G. Ferrari and G. Bertuccio, Shot Noise in Linear Macro- [5] R. Kubo, The fluctuation-dissipation theorem, scopic Resistors, Phys. Rev. Lett. 92, 226601 (2004). Rep. Prog. Phys. 29, 255 (1966).