The Thermodynamics of Nonlinear Bolometers Near Equilibrium

Transcription

1 TES III, Gainesville The Thermodynamics of Nonlinear Bolometers Near Equilibrium Kent Irwin James Beall Randy Doriese William Duncan Lisa Ferreira Gene Hilton Rob Horansky Ben Mates Nathan Miller Galen O Neil Carl Reintsema Dan Schmidt Joel Ullom Leila Vale Yizi Xu Barry Zink Paper:

2 Mather s ansatz The nonequilibrium part of Mather s theory is based on an ansatz Callen & Welton s classical generalization of Nyquist s formula: Mather realized that Callen & Welton can give nonsensical results for bolometers (including negative noise). Intuitively, he replaced it with an ansatz: Model electrical noise as a Gaussian Johnson voltage noise source in series with bolometer with noise PSD based on total resistance Work done by or on Johnson noise is coupled to thermal system. ( ) S 4 k T = Re Z ω V B (non-markov, rigorous in equilibrium) S = 4k TR V B R V I Mather s unstated ansatz has been very useful, but it is not rigorous

3 Nonequilibrium thermodynamics Existing (e.g. Mather s) first-principles calculations of thermodynamic noise in resistive bolometers assume: linear dissipative elements equilibrium Johnson voltage noise Gaussian noise sources In contrast, the following solution is a thermodynamically correct calculation of noise that is rigorous for: a simple Markovian bolometer (no hidden variables) with linear and quadratically nonlinear R and G deviating from equilibrium to 1 st and 2 nd order

4 A closed thermodynamic model We solve a thermodynamically closed model with a steady-state nonequilibrium solution: a voltage bias provided by an infinite capacitor, or a current bias provided by an infinite inductor. Total energy is constant Entropy always increases The microscopic physical processes are time reversible U(T) dq/dt dj/dt Bolometer thermal energy Heat flowing to the bath Joule power dissipation Constant current bias (e.g. thermistor) dq/dt Thermal circuit U(T) G T 0 dj/dt Electrical circuit R + - I C L 8 dq/dt Constant voltage bias (e.g. TES) Thermal circuit c(t)=du(t)/dt U(T) dj/dt G(δT) T 0 Electrical circuit Φ=LI L R(I,δT) I + - q=cv ext C 8

5 Nonequilibrium thermodynamics Equilibrium thermodynamics I δt δt = 0 = 0 T T 0 In equilibrium, Mather s bolometer theory is rigorous Electrical noise is Johnson noise Thermal noise is phonon noise or TFN thermal fluctuation noise S S V P = 4k TR = 4 B k T2G B (One-sided power spectral density (PSD)) 1 st and 2 nd -order nonequilibrium thermodynamics Third-order deviations dropped I 3 δt I δt 0 etc. Mather s bolometer theory is not rigorous Electrical noise is??? Thermal noise is??? S S V P = 4k TR = 4 B k T2G B

6 Nonlinear thermodynamics Quadratic nonlinearity We solve for linear and quadratic dissipative elements (the electrical resistance and the thermal conductance). Higher orders of nonlinearity are dropped. Thermal circuit c(t)=du(t)/dt U(T) dj/dt dq/dt G(δT) T 0 Electrical circuit Φ=LI L R(I,δT) I + - q=cv ext C 8 R(I,δT) V I ρ 1 + ρ 2 I + ζδt G(δT) dq/dt δt γ 1 + γ 2 δt

7 Thermo-electric differential equations Thermal circuit c(t)=du(t)/dt U(T) dj/dt dq/dt G(δT) T 0 Electrical circuit Φ=LI L R(I,δT) I + - q=cv ext C 8 U(T) dq/dt dj/dt Bolometer thermal energy Heat flowing to the bath Joule power dissipation R(I,δT) V I ρ 1 + ρ 2 I + ζδt G(δT) dq/dt δt γ 1 + γ 2 δt The differential equations describing this system are: Kirchhoff s voltage rule Definition of current Energy conservation

8 Microcanonical ensemble In a canonical ensemble, a thermodynamic system is in equilibrium with an external heat bath, and there is one external temperature that does not fluctuate. In a microcanonical ensemble, the heat bath is internal to the system, so there can be more than one temperature in the system. Both are equivalent for a large enough system. For convenience, here we use a microcanonical ensemble. The entropy change ds of the microcanonical ensemble ( 1 ds = 1 ) dq + dj T 0 T T dj = Φ L dφ q C dq dq = dj du ( 1 ds = T 1 ) T 0 du 1 T 0 Φ L dφ 1 T 0 q C dq

9 Thermodynamic forces In a microcanonical ensemble, the thermodynamic forces are the partial derivatives of the entropy X α = S x α α =Φ,q,U The state variables x Φ Φ=LI x q q x U U The conjugate thermodynamic forces X Φ = Φ T 0 L = I T 0 X q = q T 0 C = V ext T 0 X U = 1 1 T 0 T = δt T 0 T

10 System differential equation Already derived Same equations in terms of thermodynamic forces ; 2 nd order terms dropped L parameters

11 Onsager reciprocal relations Doctoral dissertation on Onsager reciprocal relations at Norwegian Institute of Technology was deemed insufficient for a PhD. Second try at dissertation (solution of Mathieu equation) deemed incomprehensible by chemistry faculty. Left without PhD Faculty position at Johns Hopkins fired in 1928 for incomprehensible teaching. Teaching position at Brown similarly incomprehensible; fired in Hired at Yale in 1933 scandal ensued, when it was discovered that he had no PhD. Finally granted PhD in Onsager relations received the 1968 Nobel Prize in Chemistry. Ended his career and life as a Distinguished University Professor at the University of Miami. Lars Onsager Time-reversal parity of the state variables added by Casimir

12 ? Do we satisfy Onsager?

13 Do we satisfy Onsager? YES

14 Correlators and noise Noise is characterized by correlations Onefold correlators: offset values Twofold correlators: power spectral density; Gaussian noise Threefold correlators: non-gaussian noise α =Φ,q,U

15 Stratonovich s equations Stratonovich s fluctuation-dissipation relations are derived solely from these few assumptions Markovian processes Time-reversal symmetry Consistency with equilibrium thermodynamics If we drop terms higher than the quadratic, we are only rigorous near equilibrium. R. Stratonovich, Nonlinear Nonequilibrium Thermodynamics I, Springer-Verlag, 1992.

16 Stratonovich s equations Here is my edited version of Stratonovich s fluctuation dissipation relations for the correlators appropriate for this microcanonical solution, carried out to 2 nd order expansion S 1 S S 1 2S α = + + αβ, x 2 αβγ, x x 2 B αβγ, γ x x β β β γ K L L k L K = k L + L k εεεl L L αβ B α, β β, α B α β γ γ, αβ α, βγ β, αγ K = k 1 εεε L + L + L 2 αβγ B α β γ α, βγ β, αγ γ, αβ S x γ The calculation can be carried out to arbitrarily high orders of expansion, valid further from equilibrium, but Higher order calculations have dissipationally undeterminable parameters A microscopic theory is needed for accuracy further from equilibrium R. Stratonovich, Nonlinear Nonequilibrium Thermodynamics I, Springer-Verlag, 1992.

17 The full solution for the bolometer Only assumptions Markovian processes Time-reversal symmetry Consistency with equilibrium thermodynamics Drop terms higher than quadratic

18 Onefold correlators The onefold correlators describe a rigorous, nonequilibrium steady state solution as C Onefold correlators: offset values dq/dt Thermal circuit c(t)=du(t)/dt U(T) dj/dt G(δT) T 0 Electrical circuit Φ=LI L R(I,δT) I + - q=cv ext C 8 Charge correlator is offset current Flux correlator is offset voltage Energy correlator is power

19 Onefold correlators The onefold correlators describe a rigorous, nonequilibrium steady state solution as C Onefold correlators: offset values dq/dt Thermal circuit c(t)=du(t)/dt U(T) dj/dt G(δT) T 0 Electrical circuit Φ=LI L + q=cv ext R(I,δT) - C 8 I Charge correlator is offset current Flux correlator is offset voltage Energy correlator is power Interesting term: Prevents the rectification of noise, eliminating a perpetual motion machine

20 Twofold correlators The two correlators describe Gaussian noise dq/dt Thermal circuit c(t)=du(t)/dt U(T) dj/dt G(δT) T 0 Electrical circuit Φ=LI L R(I,δT) I + - q=cv ext C 8 Twofold correlators: noise PSD Nonequilibrium Johnson noise Nonequilibrium phonon noise Electrothermal feedback

21 Twofold correlators The two correlators describe Gaussian noise dq/dt Thermal circuit c(t)=du(t)/dt U(T) dj/dt G(δT) T 0 Electrical circuit Φ=LI L R(I,δT) I + - q=cv ext C 8 Twofold correlators: noise PSD Nonequilibrium Johnson noise Nonequilibrium phonon noise Electrothermal feedback Equilibrium Johnson noise S = 4 V k TR B

22 Twofold correlators The two correlators describe Gaussian noise dq/dt Thermal circuit c(t)=du(t)/dt U(T) dj/dt G(δT) T 0 Electrical circuit Φ=LI L R(I,δT) I + - q=cv ext C 8 Twofold correlators: noise PSD Nonequilibrium Johnson noise Nonequilibrium phonon noise Electrothermal feedback = 4k T G Equilibrium thermal fluctuation noise 2 S P B

23 β noise S = 4k Tρ + 12k T ρ I+ 4k T ζδt V B 1 B 0 2 B 0 Note that if δt 0, this equation is consistent with Stratonovich s calculation of the noise in a nonlinear resistor without temperature dependence. Now introducing the usual alpha and beta parameters for the TES: And dropping 2 nd -order terms: S = 4k TR 1+ 2β V B I All the terms dependent on α cancel, but we are left with β noise. This calculation has a more complex result than Mather s ansatz.

24 Thermal fluctuation noise K 2 k T = γ T + γ T δ T UU B S = 4 k T γ T + γ T δ T P B Our equation for the thermal fluctuation noise, or phonon noise correlator, is consistent with Mather s expressions for phonon noise in a diffuse link. It is also consistent with some expressions for the limit of a specular link.

25 Threefold correlator There is a threefold correlator K 12 k B T ρ ΦΦΦ = The noise is (very) weakly non-gaussian This isn t in the equilibrium theory.

26 Conclusions Mather s classic bolometer theory is a useful, but imperfect ansatz. Nonequilibrium thermodynamics provides a rigorous expression for the noise of a simple nonlinear bolometer fairly near to equilibrium. Fairly is difficult to define. The noise is weakly non-gaussian There is a β-noise contribution from nonlinearity close to equilibrium: S 4k TR = 1+ 2β V B I Far from equilibrium, the phrase excess noise is misleading. There is just noise and its ratio to Mather s (imperfect) ansatz. Simple thermodynamics has no prediction for the baseline noise far from equilibrium Paper: