Duality and reciprocity of fluctuation-dissipation relations in conductors Lino Reggiani∗ Dipartimento di Matematica e Fisica, “Ennio de Giorgi”, Universit`adel Salento, via Monteroni, I-73100 Lecce, Italy and CNISM, Via della Vasca Navale, 84 - 00146 Roma, Italy Eleonora Alfinito Dipartimento di Ingegneria dell’ Innovazione, Universit`a del Salento, via Monteroni, I-73100 Lecce, Italy and CNISM, Via della Vasca Navale, 84 - 00146 Roma, Italy Tilmann Kuhn Institut f¨ur Festk¨orpertheorie, Universit¨at M¨unster, Wilhelm-Klemm-Str. 10, 48149 M¨unster, Germany (Dated: October 9, 2018) By analogy with linear-response we formulate the duality and reciprocity properties of current and voltage fluctuations expressed by Nyquist relations including the intrinsic bandwidths of the respective fluctuations. For this purpose we individuate total-number and drift-velocity fluctuations of carriers inside a conductor as the microscopic sources of noise. The spectral densities at low frequency of the current and voltage fluctuations and the respective conductance and resistance are related in a mutual exclusive way to the corresponding noise-source. The macroscopic variance of current and voltage fluctuations are found to display a dual property via a plasma conductance that admits a reciprocal plasma resistance. Analogously, the microscopic noise-sources are found to obey a dual property and a reciprocity relation. The formulation is carried out in the frame of the grand canonical (for current noise) and canonical (for voltage noise) ensembles and results are derived which are valid for classical as well as for degenerate statistics including fractional exclusion statistics. The unifying theory so developed sheds new light on the microscopic interpretation of dissipation and fluctuation phenomena in conductors. In particular it is proven that, as a consequence of the Pauli principle, for Fermions non-vanishing single-carrier velocity fluctuations at zero temperature are responsible for diffusion but not for current noise, which vanishes in this limit. PACS numbers: 05.40.-a: 05.40.Ca; 72.70.+m Introduction - The dual property in electrical transport in the outside circuit. The individuation of the noise in the linear-response regime asserts that perturbation sources at a kinetic level, and thus beyond the simple (applied voltage V or imposed current I) and response temperature model, is a major issue in statistical physics (measured current or voltage drop) can be interchanged that received only partial, and sometimes controversial, with the associated kinetic coefficients (conductance G answers even in the basic literature [2–6]. Ultimately, a or resistance R, respectively), being reciprocally inter- unifying theory that applies generally to classical and de- related. According to Ohm’s law, for a homogeneous generate statistics, including explicitly Fermi-Dirac and conductor the dual property gives Bose-Einstein distribution functions, is not available to our knowledge. Here, all these issues will be addressed I = GV and V = RI, (1) and formally solved in the framework of the basic laws of statistical mechanics. with the reciprocity relation given by For the analysis of current or voltage fluctuations on a GR =1 . (2) kinetic level a correct system definition becomes of prime arXiv:1606.01659v1 [cond-mat.stat-mech] 6 Jun 2016 importance. On the one hand, the microscopic model The most used applications of the above properties are for carrier transport implies a well-defined equivalent cir- Th´evenin’s and Norton’s theorems that in electrotech- cuit. On the other hand, the measurement of current or nics are two equally valid methods of reducing a complex voltage fluctuations in the outside circuit is reflected in linear-network down to something simpler to analyze [1]. the boundary conditions for the microscopic modeling, The aim of this letter is to formulate an analogous which determine the choice of the appropriate statisti- dual property and reciprocity relation for electrical fluc- cal ensemble. Current noise is measured in the outside tuations at thermal equilibrium. Here the perturbation short-circuit, which implies an open system where carri- is the microscopic source of spontaneous fluctuations in- ers may enter or leave the sample, thus referring to the side a conductor (taken as the physical system), and the grand canonical ensemble (GCE). Voltage noise is mea- response is the variance of the macroscopic response (i.e., sured in the outside open circuit when the carrier num- the variance of current or voltage fluctuations) measured ber in the sample is fixed, thus referring to the canonical 2 ensemble (CE). While it is well-known that in the ther- modynamic 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 consid- ered. 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 ther- mal 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.