Large deviations of the Lyapunov exponent in 2D matrix Langevin dynamics with applications to one-dimensional Anderson Localization models
LLarge deviations of the Lyapunov exponent in 2D matrix Langevin dynamicswith applications to one-dimensional Anderson Localization models
C´ecile Monthus
Institut de Physique Th´eorique, Universit´e Paris Saclay, CNRS, CEA, 91191 Gif-sur-Yvette, France
For the 2D matrix Langevin dynamics that corresponds to the continuous-time limit of the productof some 2 × I. INTRODUCTION
Products of random matrices and their continuous counterparts play a major role in Probability Theory andin Statistical Physics, with important applications in the field of random dynamical systems and in the field ofdisordered systems, either classical or quantum (see [1–10] and references therein). The Lyapunov exponent thatmeasures their exponential growth is an essential observable. Since its typical value appears with probability one inthe thermodynamical limit, the main goal has been usually to compute this typical value in various models via theDyson-Schmidt invariant measure method. However, more recent works have analyzed in great detail the finite-sizefluctuations of the Lyapunov exponent via its first cumulants or via its full large deviations properties, either fordynamical models with noise [11–13], for Anderson Localization models [14–16], and for products of random matrices[17], in particular for random matrices of the group SL (2 , R ) [18–20] that are related to various Localization models.In the present paper, we consider the 2D matrix Langevin dynamics that corresponds to the continuous-time limitof the product of some 2 × a r X i v : . [ c ond - m a t . d i s - nn ] J a n at level 2.5 of the Riccati process. In section V, the link with the alternative analysis in terms of the tilted dynamicsand of the conditioned dynamics is discussed in detail in order to obtain another interesting perspective. Finally, weapply this large deviation analysis to the Lyapunov exponent in one-dimensional Anderson localization models withrandom scalar potential (section VI) and with random supersymmetric potential (section VII). Our conclusions aresummarized in section VIII. In the appendices, the Euler-Lagrange optimization procedure described in section IV issolved perturbatively to obtain explicitly the first cumulants, both when the Riccati steady state is a non-equilibriumstate with current (Appendix A) and when the steady state is an equilibrium state without current (Appendix B). II. LYAPUNOV EXPONENT FOR TWO-DIMENSIONAL MATRIX LANGEVIN DYNAMICS
In this section, we introduce the notions and the notations that will be useful in the whole paper. The two-dimensional matrix Langevin dynamics is defined in cartesian coordinates in subsection II A, translated into polarcoordinates in subsection II B, while the Riccati variable is introduced in subsection II C. Then we discuss the finite-time Lyapunov exponent λ T in subsection II D, the other Lyapunov exponent in subsection II E, and the density ofzeroes of the first cartesian component in subsection II F. Finally, the notations for the large deviation analysis ofthe finite-time Lyapunov exponent λ T are introduced in subsection II G, with the rate function I ( λ ) and the scaledcumulant generating function µ ( k ), while their possible Gallavotti-Cohen symmetry is discussed in subsection II H. A. Cartesian coordinates ( y ( t ) , y ( t )) We consider the following Langevin dynamics for the two-dimensional vector of real components y ( t ) and y ( t ) ddt (cid:18) y ( t ) y ( t ) (cid:19) = ( M + η ( t ) W ) (cid:18) y ( t ) y ( t ) (cid:19) = (cid:18) M + η ( t ) W M + η ( t ) W M + η ( t ) W M + η ( t ) W (cid:19) (cid:18) y ( t ) y ( t ) (cid:19) (1)where M and W are two given 2 × η ( t ) is a Gaussian white noise < η ( t ) > = 0 < η ( t ) η ( t (cid:48) ) > = δ ( t − t (cid:48) ) (2)Since the noise η ( t ) is multiplicative in the stochastic differential Eq. 1, we need to specify that we will use theStratonovich interpretation.Eq. 1 is interesting on its own as a first-order linear dynamical system perturbed by noise. It can also appearas a reformulation of second-order differential equations in one dimension (see sections VI and VII on Andersonlocalization models). Finally, Eq. 1 corresponds to the continuous-time limit of the product of some 2 × p =1 , ,..,n (cid:18) y ( n ∆ t ) y ( n ∆ t ) (cid:19) = Ω n (cid:18) y (( n − t ) y (( n − t ) (cid:19) = Ω n Ω n − ... Ω Ω (cid:18) y (0) y (0) (cid:19) (3) B. Polar coordinates ( e ξ ( t ) , θ ( t )) Via the change of variables towards polar coordinates with modulus e ξ ( t ) = (cid:112) y ( t ) + y ( t ) and angle θ ( t ) y ( t ) + iy ( t ) ≡ e ξ ( t )+ iθ ( t ) = e ξ ( t ) (cos θ ( t ) + i sin θ ( t ))˙ y ( t ) + i ˙ y ( t ) = [ ˙ ξ ( t ) + i ˙ θ ( t )] e ξ ( t )+ iθ ( t ) (4)one obtains that the polar angle θ ( t ) evolves according to the following Langevin equation independent of ξ ( t )˙ θ ( t ) = (cid:2) M cos θ ( t ) − M sin θ ( t ) + ( M − M ) cos θ ( t ) sin θ ( t ) (cid:3) + η ( t ) (cid:2) W cos θ ( t ) − W sin θ ( t ) + ( W − W ) cos θ ( t ) sin θ ( t ) (cid:3) (5)while the Langevin equation for ξ ( t ) involves only the angle θ ( t ) and the noise η ( t ) on the right hand-side˙ ξ ( t ) = (cid:2) M cos θ ( t ) + M sin θ ( t ) + ( M + M ) cos θ ( t ) sin θ ( t ) (cid:3) + η ( t ) (cid:2) W cos θ ( t ) + W sin θ ( t ) + ( W + W ) cos θ ( t ) sin θ ( t ) (cid:3) (6)The two right-hand sides of Eqs 5 and 6 can be written in terms of the double angle (2 θ ( t )) only, and it is thusconvenient to use instead the Riccati variable as we now recall. C. Langevin dynamics for the Riccati variable R ( t ) = tan θ ( t ) = y ( t ) y ( t ) Although the angle θ ( t ) has the advantage to be periodic on the finite ring [ − π , π ] without singularities, one usuallyprefers to work instead with the Riccati variable R ( t ) ≡ tan θ ( t ) = y ( t ) y ( t ) (7)that has the disadvantages to live on the periodic infinite ring ] − ∞ , + ∞ [ and to become infinite R = ∞ when thefirst coordinate vanishes y ( t ) = 0, because its Langevin dynamics˙ R ( t ) = a [ R ( t )] + η ( t ) b [ R ( t )] (8)involves two functions that are simply polynomials of degree 2 in Ra [ R ] ≡ M + ( M − M ) R − M R b [ R ] ≡ W + ( W − W ) R − W R (9)It is thus technically simpler to work with the Riccati variable R , but one should always keep in mind that the realaxis R ∈ ] − ∞ , + ∞ [ is a periodic ring where the two infinities R = ±∞ are glued together. In addition, many integralsover R that will appear in the analysis will only be convergent in the Cauchy principal value sense. Whenever one isconfused about the singularities or the physical meaning in the asymptotic regimes R → ±∞ , one can always returnto the angle interpretation to clarify the problems. D. Finite-time Lyapunov exponent λ T as an additive functional of the Riccati process R (0 ≤ t ≤ T ) The finite-time Lyapunov exponent λ T characterizes the exponential growth of the initial 2D process ( y ( t ) , y ( t ))of Eq. 1 during the time-window 0 ≤ t ≤ T . One can thus consider different definitions based either on the growthof the modulus (cid:112) y ( t ) + y ( t ) = e ξ ( t ) , or on the growth of the absolute value of the first component | y ( t ) | or on thegrowth of the absolute value of the second component | y ( t ) | λ ( modulus ) T ≡ T ln (cid:32) (cid:112) y ( T ) + y ( T ) (cid:112) y (0) + y (0) (cid:33) = ξ ( T ) − ξ (0) T = 1 T (cid:90) T dt ˙ ξ ( t ) λ ( y ) T ≡ T ln (cid:12)(cid:12)(cid:12)(cid:12) y ( T ) y (0) (cid:12)(cid:12)(cid:12)(cid:12) = 1 T (cid:90) T dt ˙ y ( t ) y ( t ) λ ( y ) T ≡ T ln (cid:12)(cid:12)(cid:12)(cid:12) y ( T ) y (0) (cid:12)(cid:12)(cid:12)(cid:12) = 1 T (cid:90) T dt ˙ y ( t ) y ( t ) (10)Since these three definitions will become equivalent for large T , one can choose the definition that is the most relevantfor the physical application under study, or the definition that is technically simpler. In the applications to AndersonLocalization models that we will consider in sections VI and VII, it is standard to consider λ ( y ) T , so in the followingwe will choose this definition λ T ≡ λ ( y ) T ≡ T ln (cid:12)(cid:12)(cid:12)(cid:12) y ( T ) y (0) (cid:12)(cid:12)(cid:12)(cid:12) = 1 T (cid:90) T dt ˙ y ( t ) y ( t ) (11)but if one prefers another definition in Eq 10 for other applications, it is straightforward to adapt the computations.Eq. 1 yields that the logarithmic growth of the first component y ( t ) reads in terms of the Riccati variable R ( t ) = y ( t ) y ( t ) ˙ y ( t ) y ( t ) = [ M + M R ( t )] + η ( t )[ W + W R ( t )] (12)It is convenient to eliminate the noise η ( t ) via the Langevin Eq 8 for the Riccati variable in order to rewrite Eq. 12 as˙ y ( t ) y ( t ) = [ M + M R ( t )] + (cid:32) ˙ R ( t ) − a [ R ( t )] b [ R ( t )] (cid:33) [ W + W R ( t )] ≡ α [ R ( t )] + ˙ R ( t ) β [ R ( t )] (13)where we have introduced the two functions α [ R ] ≡ [ M + M R ] − a [ R ] b [ R ] [ W + W R ]= ( M W − M W ) + R ( M W − M W + M W − M W ) + R ( M W − M W ) W + ( W − W ) R − W R β [ R ] ≡ W + W Rb [ R ] = W + W RW + ( W − W ) R − W R (14)Plugging Eq. 13 into Eq. 11 yields that the finite-time Lyapunov exponent λ T can be rewritten as the additivefunctional of the Riccati process R ( t ) over the time-window 0 ≤ t ≤ Tλ T = 1 T (cid:90) T dt (cid:104) α [ R ( t )] + ˙ R ( t ) β [ R ( t )] (cid:105) (15) E. The other finite-time Lyapunov exponent λ minT The finite-time Lyapunov exponent λ T discussed in the previous subsection is actually the maximum Lyapunovexponent λ T = λ maxT (16)that governs the growth of any solution of Eq. 1 considered independently. However if one considers two solutions( ± ) of Eq. 1 ddt (cid:18) y ± ( t ) y ± ( t ) (cid:19) = ( M + η ( t ) W ) (cid:18) y ± ( t ) y ± ( t ) (cid:19) (17)the determinant of these two solutions∆( t ) ≡ (cid:12)(cid:12)(cid:12)(cid:12) y +1 ( t ) y − ( t ) y +2 ( t ) y − ( t ) (cid:12)(cid:12)(cid:12)(cid:12) = y +1 ( t ) y − ( t ) − y +2 ( t ) y − ( t ) (18)allows to analyze the dynamics of the phase space volume : the exponential growth of ∆( t ) over the time-window0 ≤ t ≤ T involves the sum of the two Lyapunov exponents λ maxT + λ minT ≡ T ln (cid:12)(cid:12)(cid:12)(cid:12) ∆( T )∆(0) (cid:12)(cid:12)(cid:12)(cid:12) = 1 T (cid:90) T dt ˙∆( t )∆( t ) (19)Since the maximum Lyapunov exponent λ maxT = λ T has already been defined in Eq. 11, Eq. 19 defines the minimumLyapunov exponent λ minT . The dynamics of the determinant of Eq. 18 is governed by the trace of the total matrix( M + η ( t ) W ) involved in Eq. 1˙∆( t )∆( t ) = Trace ( M + η ( t ) W ) = ( M + M ) + η ( t )( W + W ) (20)So Eq. 19 only involves the traces of the two matrices M and W and the integral of the white noise η ( t ) over thetime-window 0 ≤ t ≤ T λ maxT + λ minT = ( M + M ) + ( W + W ) (cid:34) T (cid:90) T dtη ( t ) (cid:35) (21)In many interesting applications, in particular in the Anderson Localization models considered in sections VI andVII, the traces of the matrices M and W vanish M + M = 0 = W + W (22)so the phase space volume is conserved ˙∆( t ) = 0, and the second Lyapunov exponent λ minT is simply the opposite of λ maxT = λ T λ minT = − λ maxT = − λ T (23)For later purposes, it is useful to rewrite the two functions of Eq. 14 for the special case of vanishing traces (Eq. 22)where we can eliminate M = − M and W = − W α conserv [ R ] = ( M W − M W ) + R ( M W − M W ) + R ( M W − M W ) W − W R − W R β conserv [ R ] ≡ W + W RW − W R − W R (24) F. Example of another interesting additive functional of the Riccati process R (0 ≤ t ≤ T ) Another example of interesting observable is the density of zeros of the first component y ( t ) = e ξ ( t ) cos θ ( t ) duringthe time-window 0 ≤ t ≤ TN T ≡ T (cid:90) T dt (cid:88) t k : y ( t k )=0 δ ( t − t k ) = 1 T (cid:90) T dt (cid:88) t k :cos θ ( t k )=0 δ ( t − t k ) (25)Using the identities for the delta function δ (cos θ ( t )) = (cid:88) t k :cos θ ( t k )=0 δ ( t − t k ) | ˙ θ ( t k ) sin θ ( t k ) | = (cid:88) t k :cos θ ( t k )=0 δ ( t − t k ) | ˙ θ ( t k ) | δ (cid:18) R ( t ) (cid:19) = (cid:88) t k : R ( tk ) =0 δ ( t − t k ) (cid:12)(cid:12)(cid:12) ˙ R ( t k ) R ( t k ) (cid:12)(cid:12)(cid:12) (26)Eq. 25 can be rewritten as the following functionals of the angle θ ( t ) or of the Riccati variable R ( t ) N T = 1 T (cid:90) T dt | ˙ θ ( t ) | δ (cos θ ( t )) = 1 T (cid:90) T dt | ˙ R ( t ) | R ( t ) δ (cid:18) R ( t ) (cid:19) (27)where the presence of the absolute values | ˙ θ ( t ) | and | ˙ R ( t ) | make them different from the additive functionals of theform of Eq. 15. Writing the Langevin Eq. 5 for θ ( t ) at times t k where cos θ ( t k ) = 0 (and thus sin θ ( t k ) = 1)˙ θ ( t k ) = − M − η ( t ) W (28)or equivalently the Langevin Eq. 8 for R ( t ) at times t k where R ( t k ) = 0˙ R ( t k ) R ( t k ) = − M − η ( t ) W (29)one sees that the vanishing of the matrix element W = 0 leads to the following simplifications in Eq. 27 N [ W =0] T = | M | T (cid:90) T dt δ (cos θ ( t )) = | M | T (cid:90) T dt δ (cid:18) R ( t ) (cid:19) (30)that is of the form of Eq. 15 even if it is for the singular function α [ R ] = δ (cid:0) R (cid:1) . It turns out that the AndersonLocalization applications considered in section VI and VII will both correspond to this special case W = 0. G. Large deviations properties of the Lyapunov exponent λ T For large T , the probability P T ( λ ) to see the finite-size Lyapunov exponent λ T = λ is expected to follow the largedeviation form P T ( λ ) (cid:39) T → + ∞ e − T I ( λ ) (31)where the rate function I ( λ ) is positive I ( λ ) ≥ λ typ that will be realized with probability one in the thermodynamic limit T → + ∞ I ( λ typ ) = 0 = I (cid:48) ( λ typ ) (32)All other values λ (cid:54) = λ typ appear with a probability P T ( λ ) that is exponentially small in T in Eq. 31, but they arenevertheless important to understand the finite-size fluctuations as we now recall. The generating function of λ T canbe evaluated from Eq. 31 via the Laplace saddle-point method for large TZ T ( k ) ≡ (cid:90) dλ P T ( λ ) e T kλ (cid:39) LT → + ∞ (cid:90) dλ e T [ kλ − I ( λ )] (cid:39) T → + ∞ e T µ ( k ) (33)where the function µ ( k ) corresponds to the Legendre transform of the rate function I ( λ ) as a consequence of thesaddle-point evaluation of the integral in λ in Eq. 33 µ ( k ) = λk − I ( λ )0 = k − I (cid:48) ( λ ) (34)with the reciprocal Legendre transform I ( λ ) = λk − µ ( k )0 = λ − µ (cid:48) ( k ) (35)The function µ ( k ) is called the scaled cumulant generating function in the field of large deviations. Its power expansionin k around k = 0 where it vanishes µ ( k = 0) = 0 as a consequence of the normalization in Eq. 33 µ ( k ) = + ∞ (cid:88) n =1 µ ( n ) (0) k n n ! (36)allows to evaluate the cumulants c n of the Lyapunov exponent λ T in terms of the derivative µ ( n ) (0) of order n at k = 0 c n = µ ( n ) (0) T n − (37)The first cumulant corresponds to the typical value λ typ where the rate function vanishes (Eq. 32) c = µ (cid:48) (0) = λ typ (38)The finite-size fluctuations around this typical value can be characterized by the next cumulants for n = 2 , , ...c = µ (cid:48)(cid:48) (0) Tc = µ (cid:48)(cid:48)(cid:48) (0) T c = µ (cid:48)(cid:48)(cid:48) (0) T (39)In specific models, the exact computation of the whole large deviation rate function I ( λ ) or of its Legendre transform µ ( k ) is usually not possible, but many results have been obtained in the references [11–20] already mentioned in theIntroduction, with three main goals:(1) compute explicitly the first cumulants via the perturbative analysis in k of scaled cumulant generating function µ ( k ), as described in particular in Refs [11, 14, 15, 18–20].(2) compute asymptotic forms of large deviations properties in various regions of the model parameters, as describedin particular in Refs [11–16, 18–20].(3) compute exactly µ ( k ) for specific integer values of k , as described in particular in Refs [11, 12, 16]. H. Gallavotti-Cohen symmetry of the rate function I ( λ ) and the scaled cumulant generating function µ ( k ) It turns out that symmetry relations for the large deviation rate function I ( λ ) of the following form involving someconstant K I ( λ ) = I ( − λ ) − Kλ (40)that corresponds to the following simple ratio for the probabilities P T ( ± λ ) to observe the value λ or its opposite ( − λ )(Eq. 31) P T ( λ ) P T ( − λ ) (cid:39) T → + ∞ e KT λ (41)or equivalently the translation of the symmetry of Eq. 40 for the Legendre transform µ ( k ) (Eqs 34 and 35) representingthe scaled cumulant generating function (Eq. 36) µ ( k ) = µ ( − K − k ) (42)have appeared independently in at least three different fields :(i) in the field of non-equilibrium dynamics, the symmetry relations of the form of Eqs 40 and 42 are famous underthe name ’Gallavotti-Cohen fluctuation relations’ (see [21–23, 32, 34, 84–92] and references therein).(ii) in the field of multifractality of wavefunctions at Anderson transition critical points (see the reviews [93, 94] andreferences therein), the symmetry of Eq. 40 is written for the singularity spectrum f ( α ) in dimension d as [95–101] f (2 d − α ) = f ( α ) + d − α (43)The relation with the field (i) has been discussed in [102].(iii) in the field of products of random matrices, the symmetry of Eq. 42 has been found for the case of real2 × K = 2 [17–19] (see [17] for the proof and for thegeneralization to real 2 n × n symplectic matrices with the parameter K = 2 n , as well as other properties in thegeneral case). For our present continuous-time model of Eq. 1, the condition of determinant unity translates into thevanishing trace condition already discussed around Eq. 22. As a consequence, whenever the vanishing trace conditionof Eq. 22 is satisfied, the symmetry relations of Eqs 40 and 42 are expected to hold with the value K = 2 I ( λ ) = I ( − λ ) − λµ ( k ) = µ ( − − k ) (44) III. STEADY-STATE OF THE RICCATI PROCESS R ( t ) AND TYPICAL LYAPUNOV EXPONENT
In this section, the Riccati process is analyzed via its Fokker-Planck generator in subsection III A and via theassociated quantum supersymmetric Hamiltonian in subsection III B. The corresponding steady-state discussed insubsection III C is either an equilibrium steady-state without current j st = 0 as discussed in section III D or a non-equilibrium steady-state with a finite stationary current j st (cid:54) = 0 as discussed in section III E, and allows to computethe typical Lyapunov exponent as recalled in subsection III F. A. Fokker-Planck operator for the Riccati process : force F ( R ) and diffusion coefficient D ( R ) The Stratonovich interpretation of the Langevin dynamics of Eq. 8 leads to the Fokker-Planck equation for theprobability P t ( R ) to see the Riccati value R at time t∂P t ( R ) ∂t = − ∂∂R (cid:18) F ( R ) P t ( R ) − D ( R ) ∂P t ( R ) ∂R (cid:19) ≡ F P t ( . ) (45)where the Fokker-Planck operator F involves the diffusion coefficient (see Eq 9) D ( R ) ≡ b [ R ]2 = (cid:2) Q + ( Q − Q ) R − Q R (cid:3) F ( R ) ≡ a [ R ] − b [ R ] b (cid:48) [ R ]2= M + ( M − M ) R − M R − (cid:2) Q + ( Q − Q ) R − Q R (cid:3) [( Q − Q ) − Q R ]2 (47) B. Associated quantum supersymmetric Hamiltonian
It is often useful to transform the non-symmetric generator of a dynamical process into a symmetric operator viathe appropriate similarity transformation (see the textbooks [103–105]). In the present context, it is thus convenientto introduce the effective potential U ( R ) ≡ − (cid:90) R dR (cid:48) F ( R (cid:48) ) D ( R (cid:48) ) U (cid:48) ( R ) = − F ( R ) D ( R ) (48)and to perform the change of variables P t ( R ) = e − U ( R )2 ψ t ( R ) (49)The Fokker-Planck Eq 45 for P t ( R ) is then transformed into the euclidean Schr¨odinger equation for ψ t ( R ) − ∂ψ t ( R ) ∂t = Hψ t ( R ) (50)where the quantum Hermitian Hamiltonian H = − ∂∂R D ( R ) ∂∂R + V ( R ) (51)corresponds to an effective position-dependent ’mass’ whenever the diffusion coefficient D ( R ) = m ( R ) dependsexplicitly on the Riccati variable R . The very specific structure of the scalar potential V ( R ) ≡ D ( R ) [ U (cid:48) ( R )] − D ( R ) U (cid:48)(cid:48) ( R )2 − D (cid:48) ( R ) U (cid:48) ( R )2= F ( R )4 D ( R ) + F (cid:48) ( R )2 (52)allows to factorize the Hamiltonian of Eq. 51 into the supersymmetric form (see the review on supersymmetricquantum mechanics [106] and references therein) H ≡ Q † Q (53)involving the two first-order operators Q ≡ (cid:112) D ( R ) (cid:18) ddR + U (cid:48) ( R )2 (cid:19) Q † ≡ (cid:18) − ddR + U (cid:48) ( R )2 (cid:19) (cid:112) D ( R ) (54)For later purposes, it is interesting to introduce the supersymmetric partner of the Hamiltonian of Eqs 51 and 53˘ H ≡ QQ † = − ∂∂R D ( R ) ∂∂R + ˘ V ( R ) (55)where the partner potential reads˘ V ( R ) ≡ D ( R ) [ U (cid:48) ( R )] D ( R ) U (cid:48)(cid:48) ( R )2 + [ D (cid:48) ( R )] D ( R ) − D (cid:48)(cid:48) ( R )2= F ( R )4 D ( R ) − F (cid:48) ( R )2 + F ( R ) D (cid:48) ( R )2 D ( R ) + [ D (cid:48) ( R )] D ( R ) − D (cid:48)(cid:48) ( R )2 (56)In particular, the commutator between Q † and Q corresponds to the difference between the two Hamiltonians, andthus to the difference between the two potentials[ Q † , Q ] ≡ Q † Q − QQ † = H − ˘ H = V ( R ) − ˘ V ( R )= F (cid:48) ( R ) − F ( R ) D (cid:48) ( R )2 D ( R ) − [ D (cid:48) ( R )] D ( R ) + D (cid:48)(cid:48) ( R )2 (57)In terms of the two functions a [ R ] and b [ R ] that appear in the Langevin Eq. 8 and that have been used to define thediffusion coefficient (Eq. 46) and the force (Eq. 47), this commutator reduces to[ Q † , Q ] = a (cid:48) [ R ] − a [ R ] b (cid:48) [ R ] b [ R ] (58)= ( M − M ) W − M ( W − W ) + 2( M W − M W ) R + [( M − M ) W − M ( W − W )] R W + ( W − W ) R − W R where we have used the explicit expressions of Eq. 9 in terms of the matrix elements of the initial model.For the special case of vanishing traces (Eq. 22) where we can eliminate M = − M and W = − W , thecommutator is directly related to the function α conserv [ R ] of Eq. 24[ Q † , Q ] conserv = 2( − M W + M W ) + 2( M W − M W ) R + 2( − M W + M W R W − W R − W R = − α conserv [ R ] (59) C. Steady-state ρ st ( R ) of the Fokker-Planck equation The steady-state solution ρ st ( R ) of the Fokker-Planck dynamics Eq. 450 = F ρ st ( . ) = − ∂∂R (cid:18) F ( R ) ρ st ( R ) − D ( R ) ∂ρ st ( R ) ∂R (cid:19) (60)corresponds to the right eigenvector r ( R ) of the Fokker-Planck operator F associated to the eigenvalue µ = 0. Thecorresponding left eigenvector l ( R )0 = F † l ( . ) = (cid:18) F ( R ) ρ st ( R ) − D ( R ) ∂ρ st ( R ) ∂R (cid:19) ∂∂R l ( R ) (61)is simply the constant function l ( R ) = 1 (62)as a consequence of the conservation of the total probability. Eq. 60 means that the corresponding steady-statecurrent j st ( R ) = F ( R ) ρ st ( R ) − D ( R ) ρ (cid:48) st ( R ) cannot depend on Rj st = F ( R ) ρ st ( R ) − D ( R ) ρ (cid:48) st ( R ) (63)In terms of the effective potential U ( R ) of Eq. 64, this equation for the steady state reads ρ (cid:48) st ( R ) + U (cid:48) ( R ) ρ st ( R ) = − j st D ( R ) (64)The integration constant will be determined by the requirement of periodicity on the Riccati ring where R = ±∞ are glued together. Since the density has to vanish at R → ±∞ to be normalizable, this periodicity constraint issomewhat tricky and it is clearer to first regularize the problem on a finite ring R ∈ [ R min , R max ]. (Note that if onechooses to work with the angle variable θ ( t ) instead of using the Riccati variable R ( t ) = tan θ ( t ) (Eq. 7), one needsindeed to solve the corresponding finite ring model). The general solution of Eq. 64 ρ Regst ( R ) = e − U ( R ) (cid:34) K − j st (cid:90) RR min dR (cid:48) D ( R (cid:48) ) e U ( R (cid:48) ) (cid:35) (65)involves an integration constant K that should be fixed by the requirement of periodicity ρ Regst ( R min ) = ρ Regst ( R max )leading to the equation for Ke − U ( R min ) K = e − U ( R max ) (cid:34) K − j st (cid:90) R max R min dR (cid:48) D ( R (cid:48) ) e U ( R (cid:48) ) (cid:35) (66)So one needs to distinguish two very different cases as we now recall.0 D. Case of periodic potential U ( R ) with an equilibrium steady-state without current j st = 0 When the effective potential U ( R ) is periodic on the Riccati ring0 = U ( R max ) − U ( R min ) = (cid:90) R max R min dRU (cid:48) ( R ) = − (cid:90) R max R min dR F ( R ) D ( R ) (67)i.e. in the limit R max → + ∞ and R min → −∞ , when the condition0 = (cid:90) + ∞−∞ dRU (cid:48) ( R ) = − (cid:90) + ∞−∞ dR F ( R ) D ( R ) (68)is satisfied, Eq. 66 yields that the steady state current vanishes j st = 0. So the steady state density reduces to theanalog of the Boltzmann distribution in the potential U ( R ) ρ eqst ( R ) = Ke − U ( R ) (69)where the constant K is fixed by the normalization1 = (cid:90) + ∞−∞ dRρ eqst ( R ) = K (cid:90) + ∞−∞ dRe − U ( R ) (70)to be the inverse of the partition function in the potential U ( R ).In the quantum mechanical language of Eq. 49, the groundstate wavefunction corresponding to ρ eqst ( R ) ψ eqst ( R ) = Ke − U ( R )2 (71)is annihilated by the operator Q of Eq. 54 Qψ eqst ( R ) = (cid:112) D ( R ) (cid:18) ddR + U (cid:48) ( R )2 (cid:19) Ke − U ( R )2 = 0 (72) E. Case of non-periodic potential with a non-equilibrium steady-state and a finite stationary current j st (cid:54) = 0 When the effective potential U ( R ) is not periodic on the Riccati ring0 (cid:54) = U ( R max ) − U ( R min ) = (cid:90) R max R min dRU (cid:48) ( R ) = − (cid:90) R max R min dR F ( R ) D ( R ) (73)i.e. equivalently in the limit R max → + ∞ and R min → −∞ (cid:54) = (cid:90) + ∞−∞ dRU (cid:48) ( R ) = − (cid:90) + ∞−∞ dR F ( R ) D ( R ) (74)Eq. 66 yields the value of the integration constant K , and the steady state density of Eq. 65 reads ρ neqst ( R ) ≡ ( − j st ) e − U ( R ) (cid:104) e − U ( R min ) (cid:82) RR min dR (cid:48) D ( R (cid:48) ) e U ( R (cid:48) ) + e − U ( R max ) (cid:82) R max R dR (cid:48) D ( R (cid:48) ) e U ( R (cid:48) ) (cid:105)(cid:2) e − U ( R min ) − e − U ( R max ) (cid:3) (75)The steady-state current j st is then determined by the normalization1 = (cid:90) R max R min dRρ Regst ( R )= ( − j st ) (cid:104) e − U ( R min ) (cid:82) R max R min dRe − U ( R ) (cid:82) RR min dR (cid:48) D ( R (cid:48) ) e U ( R (cid:48) ) + e − U ( R max ) (cid:82) R max R min dRe − U ( R ) (cid:82) R max R dR (cid:48) D ( R (cid:48) ) e U ( R (cid:48) ) (cid:105)(cid:2) e − U ( R min ) − e − U ( R max ) (cid:3) (76)In the limit R min → −∞ and R max → + ∞ , one then needs to take into account the behavior of the potential U ( R )for R → ±∞ to obtain the appropriate solution on the infinite Riccati ring. As an example, let us describe the casewhere the potential difference U (+ ∞ ) − U ( −∞ ) = + ∞ diverges.1
1. Non-equilibrium steady state when the potential difference diverges U (+ ∞ ) − U ( −∞ ) = + ∞ If the potential difference U (+ ∞ ) − U ( −∞ ) diverges towards + ∞ U (+ ∞ ) − U ( −∞ ) = (cid:90) + ∞−∞ dRU (cid:48) ( R ) = − (cid:90) + ∞−∞ dR F ( R ) D ( R ) = + ∞ (77)only the left terms survive in the numerator and denominator of Eq. 75 in the limit R min → −∞ and R max → + ∞ ,so the non-equilibrium solution on the infinite Riccati ring reduces to ρ neqst ( R ) = ( − j st ) e − U ( R ) (cid:90) R −∞ dR (cid:48) D ( R (cid:48) ) e U ( R (cid:48) ) (78)and its normalization determines the negative steady-state current j st <
01 = (cid:90) + ∞−∞ dRρ neqst ( R ) = ( − j st ) (cid:90) + ∞−∞ dRe − U ( R ) (cid:90) R −∞ dR (cid:48) D ( R (cid:48) ) e U ( R (cid:48) ) (79)If U (cid:48) ( R ) = − F ( R ) D ( R ) is large for R → ±∞ , the saddle-point evaluation of Eq 78 yields that the asymptotic behaviors ofthe density at R → ±∞ ρ st ( R ) (cid:39) R →±∞ − j st D ( R ) U (cid:48) ( R ) = j st F ( R ) (80)only involves the steady state current j st and the force F ( R ), i.e. the diffusion contribution becomes negligible in thedifferential Eq. 63 for R → ±∞ .
2. Meaning of the non-equilibrium solution in the quantum mechanical language
In the quantum mechanical language of Eq. 49, the right wavefunction corresponding to the non-equilibriumsolution ρ neqst ( R ) of Eq. 75 ψ [ r ] st ( R ) = e U ( R )2 ρ neqst ( R ) (81)is not annihilated by the operator Q of Eq. 54 (in contrast to Eq. 72 concerning the equilibrium case) Qψ [ r ] st ( R ) = (cid:112) D ( R ) (cid:18) ddR + U (cid:48) ( R )2 (cid:19) ψ [ r ] st ( R ) = ( − j st ) (cid:112) D ( R ) e U ( R )2 (82)but this state is annihilated by the operator Q † of Eq. 54 Q † (cid:16) Qψ [ r ] st ( R ) (cid:17) = (cid:18) − ddR + U (cid:48) ( R )2 (cid:19) (cid:112) D ( R ) (cid:32) ( − j st ) (cid:112) D ( R ) e U ( R )2 (cid:33) = 0 (83)as it should to produce the ground-state of energy zero of the Hamiltonian of Eq. 53. Using the trivial left eigenvector l ( R ) = 1 of Eq. 62, the change of variable conjugated to Eq. 81 yields that the corresponding left wavefunction stillcorresponds to the equilibrium wavefunction (Eq. 71) ψ [ l ] ( R ) = e − U ( R )2 l ( R ) = e − U ( R )2 (84)So even if the quantum Hamiltonian is Hermitian, the periodicity constraint on the Fokker-Planck right and lefteigenvectors can induce a difference between the associated left and right eigenwavefunctions.2 F. Typical Lyapunov exponent in terms of the steady-state
In the thermodynamics limit T → + ∞ , the finite-time Lyapunov exponent λ T will converge towards its typical value(Eq. 32), where the additive functional of the Riccati process R ( t ) of Eq. 15 can be evaluated from the steady-state ρ st ( R ) of the Riccati variable and the steady current j st discussed above λ typ = λ ( T → + ∞ ) = (cid:90) + ∞−∞ dRρ st ( R ) α ( R ) + j st β tot (85)with the notation β tot ≡ (cid:90) + ∞−∞ dRβ ( R ) (86)Now that we have recalled how the typical value λ typ can be computed in terms of the steady state of the Riccativariable, we will focus on its large deviations properties in the next sections. IV. ANALYSIS VIA THE LARGE DEVIATIONS AT LEVEL 2.5 OF THE RICCATI PROCESS
As recalled in the Introduction, the formulation of the Large Deviations at Level 2.5 has been a major achievementfor various Markovian dynamics, including Markov Chains (discrete-space and discrete-time) [10, 30, 37–39], MarkovJump processes (discrete-space and continuous-time) [30, 33, 34, 39–50] and Diffusion processes (continuous-spaceand continuous-time) [33, 34, 39, 43, 51, 52].In this section, the goal is to apply this Large Deviation analysis at Level 2.5 to the Riccati process over the largetime window 0 ≤ t ≤ T . The empirical density and the empirical current are introduced in subsection IV A and allowto reconstruct the finite-size exponent λ T . As a consequence, the corresponding explicit rate function at level 2.5recalled in subsection IV B can be used to analyze the generating function of the Lyapunov exponent as explained insubsection IV C. The corresponding optimization problem leads to Euler-Lagrange equations in subsection IV D and tothe scaled cumulant generating function µ ( k ) in subsection IV E. Finally, we describe how the optimization procedurecan be decomposed in two steps in section IV F, while the special case of an equilibrium steady-state requires somechanges as discussed in subsection IV G. A. Empirical density ρ T ( R ) and empirical current j T ( R ) of the Riccati process during ≤ t ≤ T The empirical density ρ T ( R ) represents the histogram of the Riccati variable R ( t ) seen during the time window0 ≤ t ≤ T ρ T ( R ) ≡ T (cid:90) T dt δ ( R ( t ) − R ) (87)with the normalization (cid:90) + ∞−∞ dRρ T ( R ) = 1 (88)The empirical current j T ( R ) measures the average of ˙ R ( t ) seen on the interval [0 , T ] when the Riccati variable R ( t )at the same time t takes the value R j T ( R ) ≡ T (cid:90) T dt ˙ R ( t ) δ ( R ( t ) − R ) (89)The empirical density ρ T ( R ) and the empirical current j T ( R ) allow to reconstruct the finite-size Lyapunov exponentof Eq. 15 λ T = 1 T (cid:90) T dt (cid:104) α [ R ( t )] + ˙ R ( t ) β [ R ( t )] (cid:105) = (cid:90) + ∞−∞ dRα ( R ) ρ T ( R ) + (cid:90) + ∞−∞ dRβ ( R ) j T ( R ) (90)or any other additive functional of this form involving other functions ( α ( R ) , β ( R )).3 B. Large deviations at level . for the empirical density ρ T ( R ) and the empirical current j T The derivative of the empirical current of Eq. 89 with respect to R vanishes as 1 /T for large T and contains onlyboundary terms corresponding to the beginning t = 0 and to the end t = T of the time-window [0 , T ] dj T ( R ) dR ≡ − T (cid:90) T dt ˙ R ( t ) δ (cid:48) ( R ( t ) − R ) = − T (cid:90) T dt ddt δ ( R ( t ) − R ) = δ ( R (0) − R ) − δ ( R ( T ) − R ) T (91)As a consequence, the empirical current j T ( R ) is independent of R for large Tj T ( R ) (cid:39) j T (92)and the large deviations at level 2.5 are formulated as follows [33, 34, 39, 43, 51, 52]. For large T , the joint probability P T [ ρ ( . ) , j ] to see the empirical density ρ T ( R ) = ρ ( R ) and the empirical current j T = j satisfy the large deviation form P T [ ρ ( . ) , j ] (cid:39) T → + ∞ δ (cid:18)(cid:90) + ∞−∞ dRρ ( R ) − (cid:19) e − T I . [ ρ ( . ) , j ] (93)where the delta function imposes the normalization constraint of Eq. 88, while the explicit rate function for theFokker-Planck dynamics of Eq. 45 I . [ ρ ( . ) , j ] = 14 (cid:90) + ∞−∞ dRD ( R ) ρ ( R ) [ j − ρ ( R ) F ( R ) + D ( R ) ρ (cid:48) ( R )] (94)vanishes only for the steady-state solution of Eq. 63. C. Generating function Z T ( k ) of the finite-size Lyapunov exponent λ T The finite-size Lyapunov exponent λ T can be rewritten in terms of the empirical density ρ T ( R ) and the empiricalcurrent j T as (Eqs 90 and 92) λ T = (cid:90) + ∞−∞ dRα ( R ) ρ T ( R ) + j T β tot (95)with the notation β tot introduced in Eq. 86. So its generating function (Eq 33) can be obtained from the jointprobability of Eq. 93 Z T ( k ) = (cid:90) dj (cid:90) D ρ ( . ) P T [ ρ ( . ) , j ] eT k (cid:20)(cid:90) + ∞−∞ dRα ( R ) ρ ( R ) + jβ tot (cid:21) (cid:39) T → + ∞ (cid:90) dj (cid:90) D ρ ( . ) δ (cid:18)(cid:90) + ∞−∞ dRρ ( R ) − (cid:19) e − T (cid:20) I . [ ρ ( . ) , j ] − k (cid:90) + ∞−∞ dR α ( R ) ρ ( R ) − kjβ tot (cid:21) (96)For large T , the evaluation via the saddle-point method means that one needs to optimize the functional in theexponential over the normalized density ρ ( R ) and over the current j . D. Optimization of the appropriate lagrangian function L k [ ρ ( . ) , ρ (cid:48) ( . ) , j ] In order to solve the optimization problem of Eq. 96, let us introduce the following Lagrangian function with theLagrange multiplier ω ( k ) associated to the normalization constraint of the density L k [ ρ ( . ) , ρ (cid:48) ( . ) , j ] ≡ (cid:90) + ∞−∞ dRD ( R ) ρ ( R ) [ j − ρ ( R ) F ( R ) + D ( R ) ρ (cid:48) ( R )] − k (cid:90) + ∞−∞ dRα ( R ) ρ ( R ) − kjβ tot + ω ( k ) (cid:18)(cid:90) + ∞−∞ dRρ ( R ) − (cid:19) (97)4The optimization with respect to the current j yields0 = ∂ L k [ ρ ( . ) , ρ (cid:48) ( . ) , j ] ∂j = (cid:90) + ∞−∞ dR D ( R ) (cid:20) j + D ( R ) ρ (cid:48) ( R ) ρ ( R ) − F ( R ) (cid:21) − kβ tot (98)Using the functional derivatives with respect to the density ρ ( R ) and with respect to its derivative ρ (cid:48) ( R ) ∂ L k [ ρ ( . ) , ρ (cid:48) ( . )] ∂ρ ( R ) = 14 D ( R ) (cid:34) F ( R ) − (cid:18) j + D ( R ) ρ (cid:48) ( R ) ρ ( R ) (cid:19) (cid:35) − kα ( R ) + ω ( k ) ∂ L k [ ρ ( . ) , ρ (cid:48) ( . )] ∂ρ (cid:48) ( R ) = 12 (cid:20) j + D ( R ) ρ (cid:48) ( R ) ρ ( R ) − F ( R ) (cid:21) (99)one obtains that the Euler-Lagrange equation for the optimization with respect to the density ρ ( R ) reads0 = ∂ L k [ ρ ( . ) , ρ (cid:48) ( . )] ∂ρ ( R ) − ddR (cid:20) ∂ L k [ ρ ( . ) , ρ (cid:48) ( . )] ∂ρ (cid:48) ( R ) (cid:21) = 14 D ( R ) (cid:34) F ( R ) − (cid:18) j + D ( R ) ρ (cid:48) ( R ) ρ ( R ) (cid:19) (cid:35) − kα ( R ) + ω ( k ) − ddR (cid:20) j + D ( R ) ρ (cid:48) ( R ) ρ ( R ) − F ( R ) (cid:21) (100)The physical interpretation is that G ( R ) ≡ j + D ( R ) ρ (cid:48) ( R ) ρ ( R ) (101)represents the effective force that would be needed to make the density ρ ( R ) and the current j typical (Eq 63). TheEuler-Lagrange Eq 100 corresponds to the following Riccati differential equation for G ( R ) G (cid:48) ( R )2 + G ( R )4 D ( R ) = F (cid:48) ( R )2 + F ( R )4 D ( R ) − kα ( R ) + ω ( k ) (102)while the optimization of Eq 98 over the current j corresponds to the condition0 = 12 (cid:90) + ∞−∞ dRD ( R ) [ G ( R ) − F ( R )] − kβ tot (103) E. Scaled cumulant generating function µ ( k ) from the optimal value of the Lagrangian With the optimal solutions for the effective force G ( R ), for the density ρ ( R ) and for the current j , one then needsto evaluate the corresponding optimal value of the Lagrangian of Eq. 97 L optk ≡ L k [ ρ ( . ) = ρ opt ( . ) , ρ (cid:48) ( . ) = ρ (cid:48) opt ( . ) , j = j opt ]= (cid:90) + ∞−∞ dR ρ ( R )4 D ( R ) (cid:20) j + D ( R ) ρ (cid:48) ( R ) ρ ( R ) − F ( R ) (cid:21) − k (cid:90) + ∞−∞ dRα ( R ) ρ ( R ) − kjβ tot = (cid:90) + ∞−∞ dRρ ( R ) (cid:32) [ G ( R ) − F ( R )] D ( R ) − kα ( R ) (cid:33) − kjβ tot (104)The Euler-Lagrange Eq. 102 can be used to replace ( − kα ( R )) in order to obtain L optk = (cid:90) + ∞−∞ dRρ ( R ) (cid:18) G ( R ) − G ( R ) F ( R )2 D ( R ) + G (cid:48) ( R ) − F (cid:48) ( R )2 − ω ( k ) (cid:19) − kjβ tot = (cid:90) + ∞−∞ dRρ ( R ) (cid:18) G ( R ) − G ( R ) F ( R )2 D ( R ) (cid:19) + (cid:90) + ∞−∞ dRρ ( R ) (cid:18) G (cid:48) ( R ) − F (cid:48) ( R )2 (cid:19) − ω ( k ) − kjβ tot (105)5The integration by parts of the second contribution can be simplified via the use of Eq 101 to replace ρ (cid:48) ( R )12 (cid:90) + ∞−∞ dRρ ( R ) ( G (cid:48) ( R ) − F (cid:48) ( R )) = − (cid:90) + ∞−∞ dRρ (cid:48) ( R ) ( G ( R ) − F ( R ))= − (cid:90) + ∞−∞ dR (cid:20) ρ ( R ) G ( R ) − jD ( R ) (cid:21) ( G ( R ) − F ( R ))= − (cid:90) + ∞−∞ dRρ ( R ) (cid:20) G ( R ) − G ( R ) F ( R )2 D ( R ) (cid:21) + j (cid:90) + ∞−∞ dR D ( R ) ( G ( R ) − F ( R )) (106)Plugging this result into Eq. 105 and using Eq. 103, leads to the simplifications L optk = − ω ( k ) + j (cid:20)(cid:90) + ∞−∞ dR D ( R ) ( G ( R ) − F ( R )) − kβ tot (cid:21) = − ω ( k ) (107)The final result is thus that the generating function of Eq. 96 displays the asymptotic behavior Z T ( k ) (cid:39) T → + ∞ e − T L optk = eT ω ( k ) (108)The identification with Eq. 33 yields that the scaled cumulant generating function µ ( k ) simply corresponds to theLagrange multiplier ω ( k ) in the optimization problem of Eq. 97 µ ( k ) = ω ( k ) (109) F. Summary of the optimization procedure in two steps
Let us now summarize how the optimization problem should be solved in two steps.
1. First step concerning the effective force G ( R ) and the scaled cumulant generating function µ ( k ) = ω ( k ) One should first compute together the effective force G ( R ) and the scaled cumulant generating function µ ( k ) = ω ( k ),by solving the first-order differential Eq. 102 for G ( R ) G (cid:48) ( R )2 + G ( R )4 D ( R ) = F (cid:48) ( R )2 + F ( R )4 D ( R ) − kα ( R ) + µ ( k ) (110)The condition of periodicity of G ( R ) on the Riccati ring will select the appropriate solution, while the condition ofEq. 103 will determine the value of the scaled cumulant generating function µ ( k ).It is now interesting to discuss in more details the physical meaning of Eq. 110. On the right handside of Eq. 110,one recognizes the supersymmetric potential of Eq. 52 associated to the force F ( R ) V [ F ( . )] ( R ) ≡ F (cid:48) ( R )2 + F ( R )4 D ( R ) (111)while the left handside of Eq. 110 represents the supersymmetric potential that would be associated to the effectiveforce G ( R ) V [ G ( . )] ( R ) ≡ G (cid:48) ( R )2 + G ( R )4 D ( R ) (112)so that Eq. 110 can be rewritten as V [ G ( . )] ( R ) = V [ F ( . )] ( R ) − kα ( R ) + µ ( k ) (113)The physical meaning is that the modified potential ( V [ F ( . )] ( R ) − kα ( R )) should be rewritten as a new supersymmetricpotential corresponding to some effective force G up to the constant µ ( k ). If one adds the kinetic term to obtain the full6quantum supersymmetric Hamiltonian associated to G , Eq. 113 yields in terms of the supersymmetric Hamiltonian H of Eqs 51 52 H [ G ( . )] ≡ − ∂∂R D ( R ) ∂∂R + V [ G ( . )] ( R ) = − ∂∂R D ( R ) ∂∂R + V [ F ( . )] ( R ) − kα ( R ) + µ ( k )= H − kα ( R ) + µ ( k ) (114)Since the supersymmetric potential H [ G ( . )] has by construction a vanishing ground-state energy, Eq. 114 means that( − µ ( k )) is ground-state energy of the Hamiltonian H k ≡ H − kα ( R ) = − ∂∂R D ( R ) ∂∂R + F (cid:48) ( R )2 + F ( R )4 D ( R ) − kα ( R ) (115)For the special case of vanishing traces (Eq. 22) where the function α conserv [ R ] is directly related to the commutatorof Eq. 59 α conserv [ R ] = − Q † Q − QQ † − H − ˘ H H ≡ Q † Q of Eq. 53 and its supersymmetricpartner ˘ H ≡ QQ † , one obtains that the Hamiltonian of Eq. 115 can be rewritten as the following linear combinationof H ≡ Q † Q and ˘ H ≡ QQ † of Eq. 55 H conservk = H − kα conserv ( R ) = (cid:18) k (cid:19) H − k H = (cid:18) k (cid:19) Q † Q − k QQ † (117)So the Gallavotti-Cohen symmetry of Eq. 44 for the ground-state energy µ ( k ) = µ ( − − k ) corresponds at the levelof the Hamiltonians H conserv − − k = − k H + (cid:18) k (cid:19) ˘ H = − k Q † Q + (cid:18) k (cid:19) QQ † (118)to an exchange of coefficients between the two Hamiltonians H ≡ Q † Q and ˘ H ≡ QQ † .
2. Second step concerning the density ρ ( R ) and the current j Once the effective force G ( R ) has been found in the first step described above, one should then compute togetherthe density ρ ( R ) and the current j by solving the first-order differential Eq. 101 for ρ ( R ) ρ (cid:48) ( R ) − G ( R ) D ( R ) ρ ( R ) = − jD ( R ) (119)This equation corresponds to the steady state Eq 63 where the force F ( R ) has been replaced by the effective force G ( R ). The condition of periodicity for ρ ( R ) on the Riccati ring will determine the appropriate solution, while thecurrent j will be fixed by the normalization of the density ρ ( R ).
3. Perturbative solution in the parameter k of the optimization procedure In order to see more explicitly how the optimization procedure described above works, its implementation at thelevel of the perturbation theory in the parameter k is described in Appendix A. G. Large deviations for a periodic potential U ( R ) with an equilibrium steady-state Up to now in this section, we have considered the case of a non-periodic potential U ( R ) with a non-equilibriumsteady-state ρ neqst and a finite stationary current j st (cid:54) = 0 (see subsection III E). In the present subsection, we focus onthe case where the potential U ( R ) is periodic on the Riccati ring and where the steady state ρ st is thus an equilibrium7steady state ρ eqst without current j st = 0 (subsection III D). We need to reconsider the whole optimization procedurein order to mention the differences :(1) The large deviations of Eqs 93 and 94 is now for the density ρ ( . ) alone P T [ ρ ( . )] (cid:39) T → + ∞ δ (cid:18)(cid:90) + ∞−∞ dRρ ( R ) − (cid:19) e − T (cid:90) + ∞−∞ dR ρ ( R ) D ( R ) (cid:20) D ( R ) ρ (cid:48) ( R ) ρ ( R ) − F ( R ) (cid:21) (120)(2) The Lagrangian of Eq. 97 only contains the density ρ ( . ) and its derivative ρ (cid:48) ( . ) L k [ ρ ( . ) , ρ (cid:48) ( . )] = 14 (cid:90) + ∞−∞ dR ρ ( R ) D ( R ) (cid:20) D ( R ) ρ (cid:48) ( R ) ρ ( R ) − F ( R ) (cid:21) − k (cid:90) + ∞−∞ dRα ( R ) ρ ( R ) + ω ( k ) (cid:18)(cid:90) + ∞−∞ dRρ ( R ) − (cid:19) (121)So the Euler-Lagrange Equation for the optimization over the density is still given by Eq. 102 G (cid:48) ( R )2 + G ( R )4 D ( R ) = F (cid:48) ( R )2 + F ( R )4 D ( R ) − kα ( R ) + ω ( k ) (122)for the effective force (Eq. 101) that does not contain the current j anymore G ( R ) ≡ D ( R ) ρ (cid:48) ( R ) ρ ( R ) (123)However the condition of Eq. 103 that had been produced by the optimization over the current is not present anymore.(3) So here, with respect to the two steps defined above in subsection IV F, one condition is missing in the firststep for G ( R ) described in subsection IV F 1, while one variable (namely the current j ) is missing in the second stepdescribed in subsection IV F 2. This means that one should consider the two steps in the opposite order. One shouldfirst solve Eq. 123 to obtain the density ρ ( R ) in terms of the effective force G ( R ). It is convenient to introduce thepotential associated to G ( R ) as in Eq. 48 U G ( R ) ≡ − (cid:90) R dR (cid:48) G ( R (cid:48) ) D ( R (cid:48) ) (124)to write the solution of Eq. 123 for the normalized density as the equilibrium solution in the effective potential U G (Eqs 69 70) ρ ( R ) = e − U G ( R ) (cid:82) + ∞−∞ dR (cid:48) e − U G ( R (cid:48) ) (125)The periodicity requirement for ρ ( R ) on the Riccati ring yields that the potential U G ( R ) should be periodic on theRiccati ring in the sense of Eq. 68 0 = (cid:90) + ∞−∞ dRU (cid:48) G ( R ) = − (cid:90) + ∞−∞ dR G ( R ) D ( R ) (126)This is the additional constraint on G ( R ) that one should take into account to solve the Euler-Lagrange Eq. 122for G ( R ) with the periodicity constraint on G ( R ) in order to be able to determine the scaled cumulant generatingfunction µ ( k ).In order to see more explicitly how this optimization procedure works, its implementation at the level of theperturbation theory in the parameter k is described in Appendix B. V. ANALYSIS VIA THE TILTED DYNAMICS OF THE RICCATI PROCESS
As recalled in the Introduction, the most standard method to analyze time-additive observables of stochasticprocesses consists in studying the appropriate ’tilted’ dynamical process [21, 26–29, 31, 34, 53, 54, 59–83] and thecorresponding ’conditioned’ process defined via the generalization of Doob’s h-transform.To the best of our knowledge, all previous studies of large deviations properties of Lyapunov exponents in variousmodels [11–20] are related to this tilted method despite different languages and different notations.8In this section, the goal is thus to describe this tilted dynamics method to analyze the large deviations propertiesof any additive functional of the form of Eq. 15 and to make the link with the approach of the previous section. Thetilted Fokker-Planck generator is introduced in subsection V A, while the associated conditioned process is discussed insubsection V B. The corresponding tilted non-Hermitian quantum operator is introduced in V C. After the descriptionsof two special simpler cases in subsections V D and V E, we return to the analysis of the general case in subsectionV F and explain the correspondence with the Euler-Lagrange optimization approach in subsection V G.
A. Eigenvalue problem for the tilted Fokker-Planck operator ˜ F k Using the the path-integral representation of the Fokker-Planck propagator of Eq. 45 (cid:104) R T | e T F | R (cid:105) = (cid:90) R ( T )= R T R (0)= R D [ R ( . )] e − (cid:90) T dt (cid:34) [ ˙ R ( t ) − F ( R ( t ))] D ( R ( t )) − [ D (cid:48) ( R ( t ))] D ( R ( t )) + D (cid:48)(cid:48) ( R ( t ))4 + F (cid:48) ( R ( t ))2 (cid:35) (127)one obtains that the generating function of Eq. 33 associated to the Lyapunov exponent given by the additivefunctional of Eq. 15 Z T ( k ) = < ek (cid:90) T dt (cid:16) α [ R ( t )] + ˙ R ( t ) β [ R ( t )] (cid:17) > (128)can be rewritten as the path-integral corresponding to the following tilted Fokker-Planck operator ˜ F k ∂ ˜ P t ( R ) ∂t = ˜ F k ˜ P t ( . ) ≡ − (cid:18) ∂∂R − kβ ( R ) (cid:19) (cid:20) F ( R ) ˜ P t ( R ) − D ( R ) (cid:18) ∂∂R − kβ ( R ) (cid:19) ˜ P t ( R ) (cid:21) + kα ( R ) ˜ P t ( R ) (129)that can be further rewritten as the continuity equation for the density ˜ P t ( R ) ∂ ˜ P t ( R ) ∂t = − ∂ ˜ J t ( R ) ∂R + ˜Σ t ( R ) (130)where the current ˜ J t ( R ) involves the tilted force [ F ( R ) + 2 kβ ( R ) D ( R )]˜ J t ( R ) = [ F ( R ) + 2 kβ ( R ) D ( R )] ˜ P t ( R ) − D ( R ) ∂ ˜ P t ( R ) ∂R (131)while the creation term˜Σ t ( R ) = (cid:2) kβ ( R ) F ( R ) + k β ( R ) D ( R ) + kβ (cid:48) ( R ) D ( R ) + kβ ( R ) D (cid:48) ( R ) + kα ( R ) (cid:3) ˜ P t ( R ) (132)describes how the density ˜ P t ( R ) can be created or destroyed. The scaled cumulant generating function µ ( k ) of Eq. 33then corresponds to the highest eigenvalue of the tilted Fokker-Planck operator ˜ F k that will dominate the propagatorfor large T (cid:104) R T | e T ˜ F k | R (cid:105) (cid:39) T → + ∞ e T µ ( k ) ˜ r k ( R T )˜ l k ( R ) (133)with the corresponding positive right eigenvector ˜ r k ( R ) µ ( k )˜ r k ( R ) = ˜ F k ˜ r k ( . ) = − (cid:18) ∂∂R − kβ ( R ) (cid:19) (cid:20) F ( R ) − D ( R ) (cid:18) ∂∂R − kβ ( R ) (cid:19)(cid:21) ˜ r k ( R ) + kα ( R )˜ r k ( R ) (134)and the corresponding positive left eigenvector ˜ l k ( R ) that is not trivial anymore (see Eq. 62) µ ( k )˜ l k ( R ) = ˜ F † k ˜ l k ( . ) = (cid:20) F ( R ) + D ( R ) (cid:18) ∂∂R + kβ ( R ) (cid:19)(cid:21) (cid:18) ∂∂R + kβ ( R ) (cid:19) ˜ l k ( R ) + kα ( R )˜ l k ( R ) (135)with the standard normalization (cid:90) + ∞−∞ dR ˜ l k ( R )˜ r k ( R ) = 1 (136)9while the periodicity condition on the Riccati ring where R = ±∞ are glued together can be imposed by firstconsidering the regularized version on the finite ring [ R min , R max ] as done previously (see Eq 65)˜ r k ( R min ) = ˜ r k ( R max )˜ l k ( R min ) = ˜ l k ( R max ) (137) B. Corresponding conditioned process constructed via the generalization of Doob’s h-transform
The probability to see the Riccati variable R at some interior time 0 (cid:28) t (cid:28) T for the tilted dynamics reads usingthe spectral asymptotic form of Eq. 133 for both time intervals [0 , t ] and [ t, T ]˜ P t ( R ) = (cid:104) R T | e ( T − t ) ˜ F k | R (cid:105)(cid:104) R | e t ˜ F k | R (cid:105) (cid:82) dR (cid:48) (cid:104) R T | e ( T − t ) ˜ F k | R (cid:48) (cid:105)(cid:104) R (cid:48) | e t ˜ F k | R (cid:105) (cid:39) (cid:28) t (cid:28) T e ( T − t ) µ ( k ) ˜ r k ( R T )˜ l k ( R ) e tµ ( k ) ˜ r k ( R )˜ l k ( R ) (cid:82) dR (cid:48) e ( T − t ) µ ( k ) ˜ r k ( R T )˜ l k ( R (cid:48) ) e tµ ( k ) ˜ r k ( R (cid:48) )˜ l k ( R ) (cid:39) (cid:28) t (cid:28) T ˜ l k ( R )˜ r k ( R ) (138)Since it is independent of the interior time t as long as 0 (cid:28) t (cid:28) T , it is useful to introduce the notation˜˜ ρ k ( R ) ≡ ˜ l k ( R )˜ r k ( R ) (139)for the stationary density of the tilted dynamics in the interior time region 0 (cid:28) t (cid:28) T , and to construct thecorresponding probability-preserving Fokker-Planck operator ˜˜ F k that has ˜˜ ρ k ( R ) = ˜˜ r k ( R ) as right eigenvector for theeigenvalue zero, while the corresponding left eigenvector ˜˜ l k ( R ) = 1 is the trivial one (Eq 62), via the generalization ofDoob’s h-transform ∂ ˜˜ P t ( R ) ∂t = ˜˜ F k ˜˜ P t ( . ) ≡ ˜ l k ( R ) ˜ F k ˜˜ P t ( . )˜ l k ( . ) − µ ( k ) ˜˜ P t ( R ) (140)Using the eigenvalue Eqs 134 and 135, one can indeed check that ˜˜ ρ k ( R ) = ˜ l k ( R )˜ r k ( R ) is the right eigenvector ofeigenvalue zero ˜˜ F k ˜˜ ρ k ( . ) = ˜ l k ( R ) ˜ F k ˜˜ ρ k ( . )˜ l k ( . ) − µ ( k )˜˜ ρ k ( R ) = ˜ l k ( R ) (cid:104) ˜ F k ˜ r k ( . ) − µ ( k )˜ r k ( R ) (cid:105) = 0 (141)and that ˜˜ l k ( R ) = 1 is the left eigenvector of eigenvalue zero˜˜ F † k ˜˜ l k ( . ) = 1˜ l k ( R ) ˜ F k ˜ l k ( . ) − µ ( k ) = 0 (142)Using the eigenvalue Eq. 135 for the left eigenvector ˜ l k ( R ), one obtains that Eq. 140 can be rewritten more concretelyas the probability-conserving Fokker-Planck Equation ∂ ˜˜ P t ( R ) ∂t = − ∂∂R (cid:34) F eff ( R ) ˜˜ P t ( R ) − D ( R ) ∂ ˜˜ P t ( R ) ∂R (cid:35) (143)where the effective force F eff ( R ) contains explicitly the left eigenvector ˜ l k ( R ) of the tilted operator ˜ F k (see Eq. 135) F eff ( R ) ≡ F ( R ) + 2 kβ ( R ) D ( R ) + 2 D ( R ) ˜ l (cid:48) k ( R )˜ l k ( R ) (144)As a consequence, the stationary density ˜˜ ρ k ( R ) = ˜ l k ( R )˜ r k ( R ) of Eq. 139 of this process is associated to the stationarycurrent ˜˜ j k = F eff ( R )˜˜ ρ k ( R ) − D ( R )˜˜ ρ (cid:48) k ( R ) (145)The physical interpretation is that ˜˜ ρ k ( R ) and ˜˜ j k ( R ) represent the density and the current conditioned to the Lyapunovexponent value λ = µ (cid:48) ( λ ) of the Legendre transform of Eq. 35.The consistency with the section IV suggests that the density ˜˜ ρ k ( R ) and the current ˜˜ j k should coincide with theoptimal density ρ ( R ) and the optimal current j of the Euler-Lagrange optimization, so that the effective force F eff ( R )should coincide with the force G ( R ) (see Eq. 101) of the section IV. We will indeed reach this conclusion at the endof the present section after some transformations.0 C. Eigenvalue problem for the tilted non-Hermitian quantum operator ˜ H k If one performs the same change of variables as in Eq. 49˜ P t ( R ) = e − U ( R )2 ˜ ψ t ( R ) (146)the tilted Fokker-Planck dynamics of Eq. 129 becomes − ∂ ˜ ψ t ( R ) ∂t = ˜ H k ˜ ψ t ( R ) (147)with the non-Hermitian Hamiltonian containing the supersymmetric potential V ( R ) introduced in Eq. 52˜ H k = − (cid:18) ∂∂R − kβ ( R ) (cid:19) D ( R ) (cid:18) ∂∂R − kβ ( R ) (cid:19) + V ( R ) − kα ( R ) (148)while its adjoint reads ˜ H † k = − (cid:18) ∂∂R + kβ ( R ) (cid:19) D ( R ) (cid:18) ∂∂R + kβ ( R ) (cid:19) + V ( R ) − kα ( R ) (149)In this language, the scaled cumulant generating function µ ( k ) of Eq. 33 corresponding to the highest eigenvalueof the tilted Fokker-Planck operator ˜ F k (Eq. 133) can be determined as follows : [ − µ ( k )] represents the smallesteigenvalue of the tilted Hamiltonian ˜ H k that will dominate the propagator for large T (cid:104) R T | e − T ˜ H k | R (cid:105) (cid:39) T → + ∞ e T µ ( k ) ˜ ψ [ r ] k ( R T ) ˜ ψ [ l ] k ( R ) (150)with the corresponding positive right and left eigenvectors − µ ( k ) ˜ ψ [ r ] k ( R ) = ˜ H k ˜ ψ [ r ] k ( R ) − µ ( k ) ˜ ψ [ l ] k ( R ) = ˜ H † k ˜ ψ [ l ] k ( R ) (151)The change of variables of Eq. 146 between the Fokker-Planck and the quantum eigenvectors˜ r k ( R ) = e − U ( R )2 ˜ ψ [ r ] k ( R )˜ l k ( R ) = e + U ( R )2 ˜ ψ [ l ] k ( R ) (152)yields that the boundary conditions of Eq. 137 become for the right and the left quantum eigenstates e − U ( Rmin )2 ˜ ψ [ r ] k ( R min ) = e − U ( Rmax )2 ˜ ψ [ r ] k ( R max ) e + U ( Rmin )2 ˜ ψ [ l ] k ( R min ) = e + U ( Rmax )2 ˜ ψ [ l ] k ( R max ) (153)In this quantum language, the stationary density of the conditioned process of Eq. 139 reads˜˜ ρ k ( R ) ≡ ˜ l k ( R )˜ r k ( R ) = ˜ ψ [ l ] k ( R ) ˜ ψ [ r ] k ( R ) (154)while the effective force of Eq. 145 becomes using U (cid:48) ( R ) = − F ( R ) D ( R ) (Eq. 48) F eff ( R ) = F ( R ) + 2 kβ ( R ) D ( R ) + 2 D ( R ) d ln ˜ l k ( R ) dR = F ( R ) + 2 kβ ( R ) D ( R ) + 2 D ( R ) (cid:34) U (cid:48) ( R )2 + d ln ˜ ψ [ l ] k ( R ) dR (cid:35) = 2 kβ ( R ) D ( R ) + 2 D ( R ) d ln ˜ ψ [ l ] k ( R ) dR (155)Before discussing the general case, it is now useful to discuss two special simpler cases.1 D. Special case β ( R ) ≡ : Hermitian Hamiltonian with a tilted scalar potential If β ( R ) ≡
0, i.e. if the additive functional of Eq. 15 contains only the first contribution involving α ( R ) λ [ β ( R ) ≡ T = 1 T (cid:90) T dtα [ R ( t )] (156)then the tilted Hamiltonian of Eq. 148˜ H [ β ( R ) ≡ k = − ∂∂R D ( R ) ∂∂R + V ( R ) − kα ( R ) = H − kα ( R ) ≡ H k (157)corresponds to the Hamiltonian H k = H − kα ( R ) already introduced in Eq. 115, that only contains the additionalcontribution ( − kα ( R )) for the scalar potential with respect to the initial Hamiltonian of Eq. 51. So this correspondsto the standard Feynman-Kac formula for functionals associated to scalar potentials [55–58]. E. Special case α ( R ) ≡ : Hermitian Hamiltonian with an electromagnetic vector potential If α ( R ) ≡
0, i.e. if the additive functional of Eq. 15 contains only the second contribution involving β ( R ) λ [ α ( R ) ≡ T = 1 T (cid:90) T dt ˙ R ( t ) β [ R ( t )] (158)then one can consider k = iq with real q in the generating function of Eq. 33 in order to generate the cumulants viathe Fourier transform Z [ α ( R ) ≡ T ( k = iq ) ≡ (cid:90) dλ P T ( λ ) e iT qλ = < eiq (cid:90) T dt ˙ R ( t ) β [ R ( t )] > (159)The tilted Hamiltonian of Eq. 148 is then Hermitian˜ H [ α ( R ) ≡ k = iq = (cid:18) − i ∂∂R − qβ ( R ) (cid:19) D ( R ) (cid:18) − i ∂∂R − qβ ( R ) (cid:19) + V ( R ) = (cid:16) ˜ H [ α ( R ) ≡ k = iq (cid:17) † (160)The tilt ( − qβ ( R )) with respect to the quantum canonical momentum operator p = − i ∂∂R can be interpreted as thepresence of the electromagnetic potential vector A ( R ) ≡ qβ ( R ) (161)This generalization of the Feynman-Kac formula to electromagnetic potential vectors has been much used in thecontext of polymer physics to take into account topological constraints [107, 108] and to analyze in detail the windingproperties of Brownian paths [57, 109–114].Since the Riccati variable R ∈ ] −∞ , + ∞ [ lives on the periodic ring, the quantum problem corresponds to the famousAharonov-Bohm effect, so the important parameter is the global phase accumulated during a lap around the ringΦ = (cid:90) + ∞−∞ dRA ( R ) = q (cid:90) + ∞−∞ dRβ ( R ) = qβ tot (162)where one recognizes the parameter β tot introduced in Eq. 86. At the level of the eigenvalue Eq. 151 µ ( iq ) ˜ ψ [ r ] iq ( R ) = ˜ H [ α ( R ) ≡ k = iq ˜ ψ [ r ] iq ( R ) (163)the fact that only the Aharonov-Bohm flux of Eq. 162 matters can be seen via the gauge transformation that redefinesthe phase of the wavefunction ˜ ψ [ r ] iq ( R ) = φ [ r ] iq ( R ) eiq (cid:90) R −∞ dR (cid:48) β ( R (cid:48) ) (164)2Eq. 163 yields that the eigenvalue equation for the new wavefunction φ [ r ] iq ( R ) involves the initial Hamiltonian H ofEq. 51 − µ ( iq ) φ [ r ] iq ( R ) = (cid:20) − ∂∂R D ( R ) ∂∂R + V ( R ) (cid:21) φ [ r ] iq ( R ) = Hφ [ r ] iq ( R ) (165)so that the parameter q will only survive in the boundary condition for the new wavefunction φ [ r ] iq ( R ) derived fromEq. 153 φ [ r ] iq ( R min ) e − U ( R min )2 = φ [ r ] iq ( R max ) e − U ( R max )2 + iqβ tot (166)via the Aharonov-Bohm flux Φ = qβ tot = q (cid:82) R max R min dRβ ( R ) of Eq. 162. F. General case with the two contributions
When the additive functional of Eq. 15 contains the two contributions in α ( R ) and β ( R ), the scalar potential V ( r )is shifted by ( − kα ( r )), while the kinetic part contains the imaginary vector potential of Eq. 161 when one returns tothe notation k = iq A ( R ) ≡ − ikβ ( R ) (167)It is still useful to perform the gauge transformations analogous to Eq. 164, even if the factors in the exponential arenot phases anymore [81]. The transformations for the right and for the left eigenvectors˜ ψ [ r ] k ( R ) = φ [ r ] k ( R ) ek (cid:90) R −∞ dR (cid:48) β ( R (cid:48) )˜ ψ [ l ] k ( R ) = φ [ l ] k ( R ) e − k (cid:90) R −∞ dR (cid:48) β ( R (cid:48) ) (168)lead to the eigenvalue equations (Eqs 151) for the new wavefunctions − µ ( k ) φ [ r ] k ( R ) = [ H − kα ( R )] φ [ r ] k ( R ) − µ ( k ) φ [ l ] k ( R ) = [ H − kα ( R )] φ [ l ] k ( R ) (169)that involve the Hermitian Hamiltonian H k = [ H − kα ( R )] of Eq. 115, while β ( R ) only survives via the globalparameter β tot of Eq. 86 in the new boundary conditions derived from Eqs 153 e − U ( Rmin )2 φ [ r ] k ( R min ) = e − U ( Rmax )2 + kβ tot φ [ r ] k ( R max ) e + U ( Rmin )2 φ [ l ] k ( R min ) = e + U ( Rmax )2 − kβ tot φ [ l ] k ( R max ) (170)In this language, the stationary density of the conditioned process of Eq. 154 reads˜˜ ρ k ( R ) = ˜ ψ [ l ] k ( R ) ˜ ψ [ r ] k ( R ) = φ [ l ] k ( R ) φ [ r ] k ( R ) (171)while the effective force of Eq. 155 becomes F eff ( R ) = 2 kβ ( R ) D ( R ) + 2 D ( R ) d ln ˜ ψ [ l ] k ( R ) dR = 2 kβ ( R ) D ( R ) + 2 D ( R ) (cid:34) − kβ ( R ) + d ln φ [ l ] k ( R ) dR (cid:35) = 2 D ( R ) d ln φ [ l ] k ( R ) dR (172)3 G. Equations for the effective force F eff ( R ) and correspondence with the Euler-Lagrange optimization Via the change of variables of Eq. 172 between φ [ l ] k ( R ) and F eff ( R ), the eigenvalue Schr¨odinger Eq. 169 for φ [ l ] k ( R ) − µ ( k ) φ [ l ] k ( R ) = [ H − kα ( R )] φ [ l ] k ( R ) = (cid:20) − ∂∂R D ( R ) ∂∂R + V ( R ) − kα ( R ) (cid:21) φ [ l ] k ( R )= − D ( R ) d φ [ l ] k ( R ) dR − D (cid:48) ( R ) dφ [ l ] k ( R ) dR + [ V ( R ) − kα ( R )] φ [ l ] k ( R ) (173)translates into the following Riccati equation for F eff ( R ) using Eq. 52 F (cid:48) eff ( R )2 + F eff ( R )4 D ( R ) = V ( R ) − kα ( R ) + µ ( k ) = F ( R )4 D ( R ) + F (cid:48) ( R )2 − kα ( R ) + µ ( k ) (174)that coincides with the Euler-Lagrange Eq. 110 for G ( R ) of the section IV.The boundary condition of Eq. 170 for φ [ l ] k ( R ) becomes for F eff ( R ) using U (cid:48) ( R ) = − F ( R ) D ( R ) (Eq. 48)0 = (cid:90) R max R min dR (cid:34) d ln φ [ l ] k ( R ) dR + U (cid:48) ( R )2 (cid:35) − kβ tot = (cid:90) R max R min dR (cid:20) F eff ( R )2 D ( R ) − F ( R )2 D ( R ) (cid:21) − kβ tot (175)that coincides with the condition of Eq. 103 for the function G ( R ) of the section IV.In summary, the effective force F eff ( R ) of the conditioned process introduced in Eq. 144 satisfies the same Riccatiequation (Eq 174), should be also periodic on the Riccati ring, and should satisfy the same boundary condition (Eq.175) as the force G ( R ) appearing in the Euler-Lagrange optimization of the section IV. So we can at last concludethat the effective force F eff ( R ) coincides with G ( R ) F eff ( R ) = G ( R ) (176)as it should by consistency between the interpretations of the two points of view. The density ˜˜ ρ k ( R ) (Eq. 139) andthe current ˜˜ j k (Eq. 145) coincide with the optimal density ρ ( R ) and the optimal current j of the Euler-Lagrangeoptimization. The perturbative solutions in k for the effective force F eff ( R ) = G ( R ), for the density ˜˜ ρ k ( R ) = ρ ( R )and the current ˜˜ j k = j are given in the Appendices A and B. VI. APPLICATION TO ANDERSON LOCALIZATION IN A RANDOM SCALAR POTENTIAL
The phenomenon of Anderson Localization [115] has attracted a continuous interest since its introduction (see thebooks [1–4] and the reviews [93, 94, 116, 117]). Among the exactly soluble one-dimensional models for the typicalLyapunov exponent (see the recent overviews [5–8] and references therein), the Halperin model [118] is based on theone-dimensional Schr¨odinger equation at energy E for the wave function ψ ( x ) Eψ ( x ) = − ψ (cid:48)(cid:48) ( x ) + v ( x ) ψ ( x ) (177)when the scalar potential v ( x ) is a gaussian white noise of strength σ (Eq. 2) v ( x ) = ση ( x ) (178)For the large deviations properties of the Lyapunov exponent, the Halperin model is also one of the most studiedmodel [11, 12, 15, 16, 18, 19] with detailed results concerning the first cumulants [11, 15, 18, 19], asymptotic forms oflarge deviations in various regions of the model parameters [12, 16, 18, 19], as well as exact results for µ ( k ) for eveninteger values of k [11, 12, 16].In this section, it is thus interesting to revisit the Halperin model in order to apply the general formalism describedin the previous sections. In subsection VI A, we recall how the second-order Schr¨odinger Eq. 177 can be recast into atwo-dimensional matrix Langevin dynamics, while the corresponding dynamics for the Riccati variable is described insubsection VI B with its corresponding Halperin steady state [118]. We then describe how the Riccati process allowsto obtain the finite-size Lyapunov exponent λ L in subsection VI C and the finite-size density of states N L in subsectionVI D. Finally, the large deviations properties of the finite-size Lyapunov exponent λ L are discussed in subsection VI E,while the iterative procedure to compute the first cumulants is given in subsection VI F.4 A. Corresponding two-dimensional matrix Langevin dynamics
Using the notations y ( x ) ≡ ψ ( x ) y ( x ) ≡ ψ (cid:48) ( x ) (179)Eq 177 can be recast into the matrix form ddx (cid:18) y ( x ) y ( x ) (cid:19) = (cid:18) ψ (cid:48) ( x ) ψ (cid:48)(cid:48) ( x ) (cid:19) = (cid:18) v ( x ) − E (cid:19) (cid:18) ψ ( x ) ψ (cid:48) ( x ) (cid:19) = (cid:18) ση ( x ) − E (cid:19) (cid:18) y ( x ) y ( x ) (cid:19) (180)One recognizes the form of Eq. 1 if the spatial coordinate x of the Anderson Localization model is identified with thetime t of the dynamical model of Eq. 1. The two matrices M and W have only a few non-vanishing matrix elements M = (cid:18) − E (cid:19) W = (cid:18) σ (cid:19) (181)In particular, the two traces vanish (Eq. 22). The conserved determinant of Eq. 18 corresponds to the wronskian oftwo independent solutions ψ ± ( x ) of the Schr¨odinger Eq. 177∆( x ) ≡ (cid:12)(cid:12)(cid:12)(cid:12) y +1 ( x ) y − ( x ) y +2 ( x ) y − ( x ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) ψ + ( x ) ψ − ( x ) ψ (cid:48) + ( x ) ψ (cid:48)− ( x ) (cid:12)(cid:12)(cid:12)(cid:12) = ψ + ( x ) ψ (cid:48)− ( x ) − ψ (cid:48) + ( x ) ψ − ( x ) (182)and the two Lyapunov exponents are opposite (Eq. 23).Finally, the vanishing trace condition of Eq. 22 yields that the symmetry relations of Eq 44 are expected to holdfor the Halperin model. B. Dynamics for the Riccati variable R ( x ) = ψ (cid:48) ( x ) ψ ( x ) The Riccati variable of Eq. 7 R ( x ) ≡ y ( x ) y ( x ) = ψ (cid:48) ( x ) ψ ( x ) (183)follows the Langevin dynamics of Eqs 8 and 9 dR ( x ) dx = − E − R ( x ) + ση ( x ) (184)So the corresponding Fokker-Planck Eq. 45 involves the constant diffusion coefficient (see Eq 46) D = σ F ( R ) = − E − R (186)The supersymmetric Hamiltonian introduced in Eqs 51 52 53 reads H = − D d dR + ( E + R ) D − R = Q † Q (187)with the operators of Eq. 54 Q ≡ √ D (cid:18) ddR + E + R D (cid:19) Q † ≡ √ D (cid:18) − ddR + E + R D (cid:19) (188)5The corresponding cubic potential of Eq. 48 U ( R ) = − (cid:90) R dR (cid:48) F ( R (cid:48) ) D = ED R + R D (189)displays the asymptotic behaviors U ( R ) (cid:39) R → + ∞ + ∞ U ( R ) (cid:39) R →−∞ −∞ (190)So the Halperin steady state [118] of the Fokker-Planck dynamics (Eq. 64) corresponds to the non-equilibrium caseof Eq. 78 ρ neqst ( R ) = ( − j st ) e − U ( R ) (cid:90) R −∞ dR (cid:48) D e U ( R (cid:48) ) = ( − j st ) e − ED R − R D (cid:90) R −∞ dR (cid:48) D e ED R (cid:48) + ( R (cid:48) )33 D (191)with the asymptotic behaviors of Eq. 80 ρ neqst ( R ) (cid:39) R →±∞ − j st D ( R ) U (cid:48) ( R ) = j st F ( R ) = − j st E + R (192)while the finite stationary current j st is fixed by the normalization1 = (cid:90) + ∞−∞ dRρ neqst ( R ) = ( − j st ) (cid:90) + ∞−∞ dRe − ED R − R D (cid:90) R −∞ dR (cid:48) D e ED R (cid:48) + ( R (cid:48) )33 D (193) C. Finite-size Lyapunov exponent λ L as an additive functional of the Riccati process R (0 ≤ x ≤ L ) The exponential growth of the absolute value of the wavefunction ψ ( x ) = y ( x ) on the interval 0 ≤ x ≤ Lλ L ≡ L ln (cid:12)(cid:12)(cid:12)(cid:12) ψ ( L ) ψ (0) (cid:12)(cid:12)(cid:12)(cid:12) = 1 L (cid:90) L dx ψ (cid:48) ( x ) ψ ( x ) = 1 L (cid:90) L dx R ( x ) (194)corresponds to the additive functional of the form of Eq. 15 with the simple functions α [ R ] = Rβ [ R ] = 0 (195)and can be thus obtained from the empirical density ρ L ( R ) of the Riccati variable (Eq. 87) via Eq. 90 λ L = (cid:90) + ∞−∞ dRRρ L ( R ) (196)In particular, its typical value (Eq. 85) can be obtained from the steady-state of Eq. 191 λ typ = (cid:90) + ∞−∞ dRRρ neqst ( R ) = ( − j st ) (cid:90) + ∞−∞ dRRe − ED R − R D (cid:90) R −∞ dR (cid:48) D e ED R (cid:48) + ( R (cid:48) )33 D (197) D. Finite-size density of states N L In Anderson Localization models, another interesting observable is the finite-size density of states of energy smallerthan E that can be obtained from the density of zeros of the solution ψ ( x ) of Eq. 177 at energy EN L ≡ L (cid:90) L dx (cid:88) x k : ψ ( x k )=0 δ ( x − x k ) (198)6Since W = 0 and M = 1 (Eq. 181), Eq. 30 yields that N L = 1 L (cid:90) L dx δ (cid:18) R ( x ) (cid:19) ≡ ˆ ρ L ( ζ = 0) (199)corresponds to the empirical density ˆ ρ L ( ζ ) at the origin ζ = 0 of the variable ζ = R on the interval 0 ≤ x ≤ L . Viathe change of variables ˆ ρ L ( ζ ) dζ = ρ L ( R ) dR , Eq. 199 yields that the finite-size density of state N L can be obtainedfrom the asymptotic behavior for R → ±∞ of the empirical density ρ L ( R ) of the Riccati variable R as N L = lim R →±∞ (cid:2) R ρ L ( R ) (cid:3) (200)In particular, its typical value from the asymptotic behavior (Eq. 80) of the steady-state of Eq. 191 N typ = lim R →±∞ (cid:2) R ρ neqst ( R ) (cid:3) = − j st (201)corresponds to the steady-state current ( − j st ). E. Large deviations for the finite-size Lyapunov exponent λ L The scaled cumulant generating function µ ( k ) of the finite-size Lyapunov exponent λ L (Eq. 33) can be analyzedvia the optimization procedure summarized in subsection IV F. Let us write the corresponding explicit equations forthe present Halperin model.The Euler-Lagrange Eq. 110 for the effective force G ( R ) reads for the present model (see Eqs 185 186 195) G (cid:48) ( R )2 + G ( R )4 D = ( E + R ) D − (1 + k ) R + µ ( k ) (202)while the condition of Eq. 103 becomes 0 = (cid:90) + ∞−∞ dR (cid:2) G ( R ) + ( E + R ) (cid:3) (203)Since the present model corresponds to the case of vanishing traces (Eq. 22) the quantum Hamiltonian of Eqs 115 H k ≡ H − kα ( R ) = − D ∂ ∂R + ( E + R ) D − (1 + k ) R (204)can be rewritten in terms of the operators Q and Q † of Eqs 188 H k = (cid:18) k (cid:19) Q † Q − k QQ † (205)as a linear combination of the initial supersymmetric Hamiltonian H = Q † Q and of its partner ˘ H = QQ † . So theGallavotti-Cohen symmetry of Eq. 44 for the ground-state energy µ ( k ) = µ ( − − k ) corresponds at the level of theHamiltonians to the exchange of coefficients as discussed in Eq. 118. F. First cumulants of the finite-size Lyapunov exponent λ L Since the present model corresponds to the case where the potential difference diverges U (+ ∞ ) − U ( −∞ ) = + ∞ , theperturbative solution described in the Appendix subsection A 3 yields that the coefficients µ m of the series expansionof µ ( k ) (Eq. A1) are given by Eq. A17 in terms of the non-equilibrium steady state ρ neqst ( R ) of Eq. 191 µ = (cid:90) + ∞−∞ dRRρ neqst ( R ) = λ typ µ = (cid:90) + ∞−∞ dR (cid:20) G ( R )4 D (cid:21) ρ neqst ( R ) µ = (cid:90) + ∞−∞ dR (cid:20) G ( R ) G ( R )2 D (cid:21) ρ neqst ( R ) µ = (cid:90) + ∞−∞ dR (cid:20) G ( R ) G ( R )2 D − G ( R )4 D (cid:21) ρ neqst ( R ) (206)7where the functions G m ( R ) are given by Eq. A16 in terms of the potential U ( R ) of Eq. 189 G ( R ) = − e U ( R ) (cid:90) + ∞ R dR (cid:48) [2 µ − R (cid:48) ] e − U ( R (cid:48) ) G ( R ) = − e U ( R ) (cid:90) + ∞ R dR (cid:48) (cid:20) µ − G ( R (cid:48) )2 D (cid:21) e − U ( R (cid:48) ) G ( R ) = − e U ( R ) (cid:90) + ∞ R dR (cid:48) (cid:20) µ − G ( R (cid:48) ) G ( R (cid:48) ) D (cid:21) e − U ( R (cid:48) ) G ( R ) = − e U ( R ) (cid:90) + ∞ R dR (cid:48) (cid:20) µ − G ( R (cid:48) ) G ( R (cid:48) ) D − G ( R (cid:48) )2 D (cid:21) e − U ( R (cid:48) ) (207)The iterative procedure goes as follows : one plugs µ = λ typ into 207 to compute G ( R ) that can be then pluggedinto Eq 206 to compute µ , that can be then plugged into Eq 207 to compute G ( R ), and so on. VII. ANDERSON LOCALIZATION IN A RANDOM SUPERSYMMETRIC POTENTIAL
In the field of exactly soluble one-dimensional Anderson Localization models for the typical Lyapunov exponent,another much studied model (besides the case of the random scalar potential considered in the previous section)involves a random supersymmetric potential [119–123]: one considers the Schr¨odinger equation at energy E for thesupersymmetric Hamiltonian Eψ ( x ) = − ψ (cid:48)(cid:48) ( x ) + (cid:2) w ( x ) + w (cid:48) ( x ) (cid:3) ψ ( x ) = (cid:18) ddx + w ( x ) (cid:19) (cid:18) − ddx + w ( x ) (cid:19) ψ ( x ) (208)where w ( x ) involves the white noise η ( x ) of strength gw ( x ) = νg + gη ( x ) (209)while the parameter ν ≥ < w ( x ) > = νg and replaces the standard notation µ ofthe literature on the Anderson Localization model of Eq 208 because the notation µ ( k ) is already used in the presentarticle for the scaled cumulant generating function of Eq. 33.For the large deviations properties of the Lyapunov exponent, many results have been obtained [15, 18] in particularfor the first cumulants and for asymptotic forms of large deviations in various regions of the model parameters.In this section, we thus revisit this supersymmetric Anderson Localization model in order to apply the generalformalism described in the previous sections. We mention the reformulation as a two-dimensional matrix Langevindynamics in subsection VII A, and the corresponding Riccati dynamics in subsection VII B. We then describe how theRiccati process allows to obtain the finite-size density of states N L in subsection VII C and the finite-size Lyapunovexponent λ L in subsection VII D. The steady state of the Riccati process is given in subsection VII E for the region ofnegative energy and in subsection VII F for the region of positive energy. Finally, we describe how the large deviationsproperties of the finite-size Lyapunov exponent λ L can be analyzed in subsection VII G, while the iterative procedureto compute the first cumulants is given in subsection VII H for the region of positive energy and in subsection VII Ifor the region of negative energy. A. Corresponding two-dimensional matrix Langevin dynamics
Using the notations y ( x ) ≡ ψ ( x ) y ( x ) ≡ ψ (cid:48) ( x ) − w ( x ) ψ ( x ) (210)Eq 208 can be recast into the matrix form ddx (cid:18) y ( x ) y ( x ) (cid:19) = (cid:18) ψ (cid:48) ( x )( w ( x ) − E ) ψ ( x ) − w ( x ) ψ (cid:48) ( x ) (cid:19) = (cid:18) w ( x ) 1 − E − w ( x ) (cid:19) (cid:18) y ( x ) y ( x ) (cid:19) = (cid:18) νg + gη ( x ) 1 − E − νg − gη ( x ) (cid:19) (cid:18) y ( x ) y ( x ) (cid:19) M and W read M = (cid:18) νg − E − νg (cid:19) W = (cid:18) g − g (cid:19) (211)Again, the two traces vanish (Eq. 22), the conserved determinant of Eq. 18 corresponds to the wronskian of twoindependent solutions ψ ± ( x ) of the Schr¨odinger Eq. 208∆( x ) ≡ (cid:12)(cid:12)(cid:12)(cid:12) y +1 ( x ) y − ( x ) y +2 ( x ) y − ( x ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) ψ + ( x ) ψ − ( x ) ψ (cid:48) + ( x ) − w ( x ) ψ + ( x ) ψ (cid:48)− ( x ) − w ( x ) ψ − ( x ) (cid:12)(cid:12)(cid:12)(cid:12) = ψ + ( x ) ψ (cid:48)− ( x ) − ψ (cid:48) + ( x ) ψ − ( x ) (212)and the two Lyapunov exponents are opposite (Eq. 23).Finally, the vanishing trace condition of Eq. 22 yields that the symmetry relations of Eq 44 are also expected tohold for the present model. B. Dynamics for the Riccati variable R ( x ) = ψ (cid:48) ( x ) ψ ( x ) − w ( x ) The Riccati variable of Eq. 7 R ( x ) ≡ y ( x ) y ( x ) = ψ (cid:48) ( x ) ψ ( x ) − w ( x ) (213)satisfies the Langevin dynamics (see Eqs 8 and 9) dRdx = − E − R ( x ) − R ( x ) w ( x ) = − E − νg R ( x ) − R ( x ) − gR ( x ) η ( x ) (214)So the corresponding Fokker-Planck Eq. 45 involves the quadratic diffusion coefficient (see Eq 46) D ( R ) = 2 g R (215)and the quadratic force (see Eq 47) F ( R ) = − E − g ( ν + 1) R − R (216)The supersymmetric Hamiltonian of Eq 51 52 reads H = Q † Q = − ∂∂R (2 g R ) ∂∂R + (cid:2) E + 2 g ( ν + 1) R + R (cid:3) g R − g ( ν + 1) − R (217)Since the derivative U (cid:48) ( R ) of Eq. 48 U (cid:48) ( R ) = − F ( R ) D ( R ) = E g R + (1 + ν ) R + 12 g (218)is singular at the origin R →
0, one needs to change the arbitrary constant in the definition of the potential U ( R ) ofEq. 48. So we will choose U ( R ) ≡ − E g R + (1 + ν ) ln | R | + R g (219)Since the signs of the leading singularities for R → ± depend on the sign of the energy E , the global structure ofthe potential U ( R ) will be completely different for positive energies E >
E <
0. Thiscomplete change in behavior is of course natural for the density of states as we now recall.9
C. Finite-size density of states N L Since one has again W = 0 and M = 1, the analysis of the finite-size density of states N L is the same as in theprevious section (Eqs 198 199) with the same final result (Eq. 200) N L = lim R →±∞ (cid:2) R ρ L ( R ) (cid:3) (220)Here from the factorized structure of the supersymmetric Hamiltonian of Eq. 208, one knows that the density ofstates vanishes in the whole region of negative energies, while it will be finite for positive energy E > N [ E< L = 0 N [ E> L > D. Finite-size Lyapunov exponent λ L Let us first recall that at zero energy E = 0, the ground-state solution of Eq. 208 ψ [ E =0] ( x ) = e (cid:82) x dyw ( y ) = e νg x + g (cid:82) x dyη ( y ) (222)involves the finite-size Lyapunov exponent λ [ E =0] L ≡ L ln (cid:12)(cid:12)(cid:12)(cid:12) ψ [ E =0] ( L ) ψ [ E =0] (0) (cid:12)(cid:12)(cid:12)(cid:12) = 1 L (cid:90) L dyw ( y ) = νg + gL (cid:90) L dyη ( y ) (223)Since it depends only on the integral of the noise η ( y ) over the region 0 ≤ y ≤ L , λ [ E =0] L is simply Gaussian aroundits typical value ( νg ).Let us now turn to the other cases with non-zero energy E (cid:54) = 0. The exponential growth of the absolute value ofthe wavefunction ψ ( x ) = y ( x ) on the interval 0 ≤ x ≤ L is measured by the finite-size Lyapunov exponent λ L ≡ L ln (cid:12)(cid:12)(cid:12)(cid:12) ψ ( L ) ψ (0) (cid:12)(cid:12)(cid:12)(cid:12) = 1 L (cid:90) L dx ψ (cid:48) ( x ) ψ ( x ) = 1 L (cid:90) L dx [ R ( x ) + w ( x )] = 1 L (cid:90) L dx (cid:20) R ( x ) − R ( x ) + E + R (cid:48) ( x )2 R ( x ) (cid:21) = 1 L (cid:90) L dx (cid:20) R ( x )2 − E R ( x ) − R (cid:48) ( x )2 R ( x ) (cid:21) ≡ L (cid:90) L dx [ α [ R ( x )] + R (cid:48) ( x ) β [ R ( x )]] (224)that corresponds to the general form of Eq. 15 with the two functions α [ R ] = R − E Rβ [ R ] = − R (225)The important parameter β tot of Eq. 86 β tot ≡ (cid:90) + ∞−∞ dRβ [ R ] (226)thus requires to use the Cauchy principal value definition of the integral both for R → ± and for R → ±∞ . With asmall cut-off (cid:15) and a large cut-off R , one obtains − β tot = lim (cid:15) → + R→ + ∞ (cid:32)(cid:90) − (cid:15) −R dRR + (cid:90) R (cid:15) dRR (cid:33) = lim (cid:15) → + R→ + ∞ (ln | − (cid:15) | − ln | − R| + ln |R| − ln | − (cid:15) | ) = 0 (227)So β tot actually vanishes, and one needs to consider only the contribution in α [ R ] in Eq. 95 λ L = (cid:90) + ∞−∞ dRα [ R ] ρ L ( R ) (228)0 E. Steady state and typical Lyapunov exponent in the region of negative energies
E < In the region of negative energies E = −| E | <
0, the asymptotic behaviors for R → ±∞ and for R → ± of thepotential of Eq. 219 U ( R ) (cid:39) R → + ∞ + ∞ U ( R ) (cid:39) R → + + ∞ U ( R ) (cid:39) R → − −∞ U ( R ) (cid:39) R →−∞ −∞ (229)yields that the Fokker-Planck dynamics will converge towards equilibrium in the region of positive Riccati variable R ∈ ]0 , + ∞ [ (see Eq. 69) ρ eqst ( R ) = e − U ( R ) Z = R − − ν e − g ( | E | R + R ) Z for 0 < R < + ∞ (230)with the corresponding partition function Z = (cid:90) + ∞ dRe − U ( R ) = (cid:90) + ∞ dRR ν e − g ( | E | R + R ) = 2 | E | − ν K ν (cid:32) (cid:112) | E | g (cid:33) (231)in terms of the Bessel function K ν ( z ). If the initial condition happens to be on the negative side R <
0, the particlewill flow towards U = −∞ and will be reinjected at U = + ∞ on the positive side R >
E < N typ = lim R →±∞ (cid:2) R ρ eqst ( R ) (cid:3) = 0 (232)as already discussed around Eq. 221.The typical value of the Lyapunov exponent of Eq. 228 λ typ = (cid:90) + ∞ dRα [ R ] ρ eqst ( R ) (233)involves the function of Eq. 225 that reads in the region of negative energy E = −| E | < α [ R ] = R − E R = 12 (cid:18) | E | R + R (cid:19) (234)One recognizes the combination (cid:16) | E | R + R (cid:17) that appear in the exponential part of the integral defining the partitionfunction of Eq. 231. As a consequence, the derivative of the partition function Z with respect to the variable ( g )corresponds to the integral − ∂ Z ∂ ( g ) = (cid:90) + ∞ dRR ν (cid:18) | E | R + R (cid:19) e − g ( | E | R + R ) = (cid:90) + ∞ dRR ν α [ R ] e − g ( | E | R + R ) = Z (cid:90) + ∞ dRα [ R ] ρ eqst ( R )= Z λ typ (235)So the typical Lyapunov exponent reads using the Bessel function K ν ( z ) of Eq. 231 λ typ = − ∂ ln( Z ) ∂ ( g ) = − (cid:112) | E | K (cid:48) ν (cid:18) √ | E | g (cid:19) K ν (cid:18) √ | E | g (cid:19) (236)and one recovers the typical value of Eq. 223 for E → − .1 F. Steady state and typical Lyapunov exponent in the region of positive energies
E > In the region of positive energies
E >
0, the asymptotic behaviors for R → ±∞ and for R → ± of the potential ofEq. 219 U ( R ) (cid:39) R → + ∞ + ∞ U ( R ) (cid:39) R → + −∞ U ( R ) (cid:39) R → − + ∞ U ( R ) (cid:39) R →−∞ −∞ (237)will produce a non-equilibrium steady state where the particle follows the descent of the potential from U ( R = + ∞ ) =+ ∞ towards U ( R = 0 + ) = −∞ , it is then reinjected at U ( R = 0 − ) = + ∞ and follows the descent of the potentialtowards U ( R = −∞ ) = −∞ where it is reinjected at U ( R = + ∞ ) = + ∞ to begin another cycle. As for Eq. 75, onecan regularize the problem on [ R min , − (cid:15) ] ∪ [+ (cid:15), R max ] to write the appropriate periodic steady state solution on thisRing where ( R min , , R max ) are glued together as in Eq. 75, and where ( − (cid:15), + (cid:15) ) are glued together. Then one takesthe limit R min → −∞ , R max → + ∞ and (cid:15) → R <
R > ρ neqst ( R <
0) = ( − j st ) e − U ( R ) (cid:90) R −∞ dR (cid:48) D ( R (cid:48) ) e U ( R (cid:48) ) = ( − j st )2 g | R | ν e − g ( R − ER ) (cid:90) R −∞ dR (cid:48) | R (cid:48) | ν − e g ( R (cid:48) − ER (cid:48) ) ρ neqst ( R >
0) = ( − j st ) e − U ( R ) (cid:90) R dR (cid:48) D ( R (cid:48) ) e U ( R (cid:48) ) = ( − j st )2 g R ν e − g ( R − ER ) (cid:90) R dR (cid:48) ( R (cid:48) ) ν − e g ( R (cid:48) − ER (cid:48) ) (238)while the steady state current j st is determined by the normalization condition1 = (cid:90) + ∞−∞ dRρ neqst ( R ) = ( − j st ) (cid:34)(cid:90) −∞ dRe − U ( R ) (cid:90) R −∞ dR (cid:48) D ( R (cid:48) ) e U ( R (cid:48) ) + (cid:90) + ∞ dRe − U ( R ) (cid:90) R dR (cid:48) D ( R (cid:48) ) e U ( R (cid:48) ) (cid:35) (239)This solution is thus very similar to Eq. 191, except that there are now two regions R <
R > R → −∞ and for R → + ∞ , the saddle-point evaluation of the two integrals of Eq. 238 as in Eq. 80 and192 leads to the asymptotic behavior ρ neqst ( R ) (cid:39) R →±∞ ( − j st ) D ( R ) U (cid:48) ( R ) = j st F ( R ) = ( − j st ) R + 2 g ( ν + 1) R + E (240)so that the steady state current ( − j st ) directly corresponds to the typical density of states (Eq. 80) as in Eq. 201 N typ = lim R →±∞ (cid:2) R ρ neqst ( R ) (cid:3) = − j st (241)(ii) For R → − and for R → + , the saddle-point evaluation of the two integrals of Eq. 238 leads to the asymptoticbehavior ρ neqst ( R ) (cid:39) R → ± = j st F (0) = ( − j st ) E (242)The typical value of the Lyapunov exponent can be obtained from the non-equilibrium steady state ρ neqst ( R ) of Eq.238 λ typ = (cid:90) + ∞−∞ dRα [ R ] ρ neqst ( R ) = (cid:90) −∞ dR (cid:18) R − E R (cid:19) ( − j st )2 g | R | ν e − g ( R − ER ) (cid:90) R −∞ dR (cid:48) | R (cid:48) | ν − e g ( R (cid:48) − ER (cid:48) )+ (cid:90) + ∞ dR (cid:18) R − E R (cid:19) ( − j st )2 g R ν e − g ( R − ER ) (cid:90) R dR (cid:48) ( R (cid:48) ) ν − e g ( R (cid:48) − ER (cid:48) ) (243)2 G. Large deviations for the finite-size Lyapunov exponent λ L The scaled cumulant generating function µ ( k ) of the finite-size Lyapunov exponent λ L (Eq. 33) is determined bythe Euler-Lagrange Eq. 110 for the effective force G ( R ) that reads for the present model (see Eqs 215 216 225) G (cid:48) ( R )2 + G ( R )42 g R = (cid:2) E + 2 g ( ν + 1) R + R (cid:3) g R − g ( ν + 1) − R − k (cid:18) R − E R (cid:19) + µ ( k ) (244)while the condition of Eq. 103 becomes0 = (cid:90) + ∞−∞ dRR (cid:2) G ( R ) + E + 2 g ( ν + 1) R + R (cid:3) (245)Since the present model corresponds to the case of vanishing traces (Eq. 22) the quantum Hamiltonian of Eqs 115can be rewritten in terms of the operators Q and Q † of Eqs 188 H k = − ∂∂R (2 g R ) ∂∂R + (cid:2) E + 2 g ( ν + 1) R + R (cid:3) g R − g ( ν + 1) − R − k (cid:18) R − E R (cid:19) = (cid:18) k (cid:19) Q † Q − k QQ † (246)as a linear combination of the initial supersymmetric Hamiltonian H = Q † Q and of its partner ˘ H = QQ † . So theGallavotti-Cohen symmetry of Eq. 44 for the ground-state energy µ ( k ) = µ ( − − k ) corresponds at the level of theHamiltonians to the exchange of coefficients as discussed in Eq. 118. H. Explicit first cumulants of the finite-size Lyapunov exponent in the region of positive energies
E > In the region of positive energy
E >
0, the steady-state is a non-equilibrium steady state ρ neqst with a steadycurrent j st (see Eq. 238), so one can apply the perturbative procedure in two steps described in the Appendix A.To take into account the singularities of the potential U ( R ) both at R → ± and at R → ±∞ (see Eq. 237), onecan first regularize the periodic Riccati ring by considering [ R min , − (cid:15) ] ∪ [+ (cid:15), R max ] to write the appropriate periodicsolutions where ( R min , , R max ) are glued together and where ( − (cid:15), + (cid:15) ) are glued together. Then one takes the limit R min → −∞ , R max → + ∞ and (cid:15) →
1. Perturbative solution in k for the effective force G ( R ) and the scaled cumulant generating function µ ( k ) One obtains that the coefficients µ m of the expansion of Eq. A1 for the scaled cumulant generating function µ ( k )are given by µ = (cid:90) + ∞−∞ dRα ( R ) ρ neqst ( R ) = λ typ µ = (cid:90) + ∞−∞ dR (cid:20) G ( R )4 D (cid:21) ρ neqst ( R ) µ = (cid:90) + ∞−∞ dR (cid:20) G ( R ) G ( R )2 D (cid:21) ρ neqst ( R ) µ = (cid:90) + ∞−∞ dR (cid:20) G ( R ) G ( R )2 D − G ( R )4 D (cid:21) ρ neqst ( R ) (247)3while the functions G m ( R ) that appear in the perturbative expansion of the effective force G ( R ) (Eq. A1) read forthe scaled cumulant generating function µ ( k ) G ( R ) = − e U ( R ) (cid:90) B + ( R ) R dR (cid:48) [2 µ − α [ R (cid:48) ]] e − U ( R (cid:48) ) G ( R ) = − e U ( R ) (cid:90) B + ( R ) R dR (cid:48) (cid:20) µ − G ( R (cid:48) )2 D (cid:21) e − U ( R (cid:48) ) G ( R ) = − e U ( R ) (cid:90) B + ( R ) R dR (cid:48) (cid:20) µ − G ( R (cid:48) ) G ( R (cid:48) ) D (cid:21) e − U ( R (cid:48) ) G ( R ) = − e U ( R ) (cid:90) B + ( R ) R dR (cid:48) (cid:20) µ − G ( R (cid:48) ) G ( R (cid:48) ) D − G ( R (cid:48) )2 D (cid:21) e − U ( R (cid:48) ) (248)where we have introduced the following notation for the upper boundary of the integral B + ( R ) = 0 for − ∞ < R < B + ( R ) = + ∞ for 0 < R < + ∞ (249)in order to avoid the writing of two separate definitions for the two regions R <
R >
2. Perturbative solution in k for the optimal density ρ ( R ) and the optimal current j Here the normalized auxiliary function (Eq. A12) involves the same upper boundary of Eq. 249 for the integralΠ( R ) ≡ ( − j st ) e U ( R ) D ( R ) (cid:90) B + ( R ) R dR (cid:48) e − U ( R (cid:48) ) (250)while the ρ m ( R ) that appear in the perturbative expansion of the density (Eq. A8) ρ ( R ) = e − U ( R ) (cid:90) RB − ( R ) dR (cid:48) e U ( R (cid:48) ) (cid:20) G ( R (cid:48) ) ρ neqst ( R (cid:48) ) − j D ( R ) (cid:21) ρ ( R ) = e − U ( R ) (cid:90) RB − ( R ) dR (cid:48) e U ( R (cid:48) ) (cid:20) G ( R (cid:48) ) ρ neqst ( R (cid:48) ) + G ( R (cid:48) ) ρ ( R (cid:48) ) − j D ( R (cid:48) ) (cid:21) ρ ( R ) = e − U ( R ) (cid:90) RB − ( R ) dR (cid:48) e U ( R (cid:48) ) (cid:20) G ( R (cid:48) ) ρ neqst ( R (cid:48) ) + G ( R (cid:48) ) ρ ( R ) + G ( R (cid:48) ) ρ ( R (cid:48) ) − j D ( R (cid:48) ) (cid:21) ρ ( R ) = e − U ( R ) (cid:90) RB − ( R ) dR (cid:48) e U ( R (cid:48) ) (cid:20) G ( R (cid:48) ) ρ neqst ( R (cid:48) ) + G ( R (cid:48) ) ρ ( R (cid:48) ) + G ( R (cid:48) ) ρ ( R (cid:48) ) + G ( R (cid:48) ) ρ ( R (cid:48) ) − j D ( R (cid:48) ) (cid:21) (251)involve the same lower boundary for the integrals as for the non-equilibrium steady state ρ neqst of Eq. 238 B − ( R ) = −∞ for − ∞ < R < B − ( R ) = 0 for 0 < R < + ∞ (252)Finally, the coefficients j m that appear in the perturbative expansion of the current (Eq. A8) are given by j = (cid:90) + ∞−∞ dR [ G ( R ) ρ neqst ( R )] Π( R ) j = (cid:90) + ∞−∞ dR [ G ( R ) ρ neqst ( R ) + G ( R ) ρ ( R )] Π( R ) j = (cid:90) + ∞−∞ dR (cid:48) [ G ( R ) ρ neqst ( R ) + G ( R ) ρ ( R ) + G ( R ) ρ ( R ) − ] Π( R ) j = (cid:90) + ∞−∞ dR [ G ( R ) ρ neqst ( R ) + G ( R ) ρ ( R ) + G ( R ) ρ ( R ) + G ( R ) ρ ( R )] Π( R ) (253)4 I. Explicit first cumulants of the finite-size Lyapunov exponent in the region of negative energies
E < In the region of negative energies
E < ρ eqst ( R ) on 0 < R < + ∞ (see Eq. 230), one cannot apply the procedure described in the subsection IV G and in the Appendix B that weremeant for an equilibrium on the periodic Riccati ring. Here the equilibrium concerns an interval 0 < R < + ∞ withtwo boundaries at R = 0 and R = + ∞ . As a consequence, one needs to reconsider the perturbative procedure andone obtains that it can be decomposed in two steps as follows.
1. Perturbative solution in k for the effective force G ( R ) and the scaled cumulant generating function µ ( k ) The general solutions of Eq. A2 for the functions G m ( R ) that appear in the series expansion of Eq. A1 for G ( R )involves some integration constant K m G m ( R ) = e U ( R ) (cid:34) K m + (cid:90) R dR (cid:48) Ω m ( R (cid:48) ) e − U ( R (cid:48) ) (cid:35) (254)In order to avoid the strong divergence as e U ( R ) for R →
0, one needs to select the value K m = 0 (255)Then in order to avoid the strong divergence as e U ( R ) for R → + ∞ , one needs to impose the vanishing of the integral0 = (cid:90) + ∞ dR (cid:48) Ω m ( R (cid:48) ) e − U ( R (cid:48) ) (256)So the solution of Eq. 254 becomes G m ( R ) = e U ( R ) (cid:90) R dR (cid:48) Ω m ( R (cid:48) ) e − U ( R (cid:48) ) = − e U ( R ) (cid:90) + ∞ R dR (cid:48) Ω m ( R (cid:48) ) e − U ( R (cid:48) ) (257)i.e. more explicitly for m = 1 , , , m ( R ) G ( R ) = e U ( R ) (cid:90) R dR (cid:48) e − U ( R (cid:48) ) [2 µ − α ( R (cid:48) )] G ( R ) = e U ( R ) (cid:90) R dR (cid:48) e − U ( R (cid:48) ) (cid:20) µ − G ( R (cid:48) )2 D ( R (cid:48) ) (cid:21) G ( R ) = e U ( R ) (cid:90) R dR (cid:48) e − U ( R (cid:48) ) (cid:20) µ − G ( R (cid:48) ) G ( R (cid:48) ) D ( R (cid:48) ) (cid:21) G ( R ) = e U ( R ) (cid:90) R dR (cid:48) e − U ( R (cid:48) ) (cid:20) µ − G ( R (cid:48) ) G ( R (cid:48) ) D ( R (cid:48) ) − G ( R (cid:48) )2 D ( R (cid:48) ) (cid:21) (258)The condition of Eq. 256 can be rewritten in terms of ρ eqst ( R ) Eq. 2300 = (cid:90) + ∞ dR Ω m ( R ) ρ eqst ( R ) (259)With the explicit expressions of Eq. A2 for Ω m ( R ), one obtains that the coefficients µ m of the perturbative expansionof Eq. A1 for the scaled cumulant generating function µ ( k ) are given by µ = (cid:90) + ∞ dRα ( R ) ρ eqst ( R ) µ = (cid:90) + ∞ dR G ( R )4 D ( R ) ρ eqst ( R ) µ = (cid:90) + ∞ dR G ( R ) G ( R )2 D ( R ) ρ eqst ( R ) µ = (cid:90) + ∞ dR (cid:20) G ( R ) G ( R )2 D ( R ) − G ( R )4 D ( R ) (cid:21) ρ eqst ( R ) (260)5
2. Perturbative solution in k for the optimal density ρ ( R ) Even if the perturbative solution for the scaled cumulant generating function µ ( k ) has already been obtained in Eq.260, it is nevertheless interesting to derive the corresponding perturbative solution in k for the optimal density ρ ( R ).The general solution of Eq. A9 for the functions ρ m ( R ) that appear in the series expansion of Eq. A8 for ρ ( R )involves some integration constant C m ρ m ( R ) = e − U ( R ) (cid:34) C m + (cid:90) R dR (cid:48) Υ m ( R (cid:48) ) e U ( R (cid:48) ) (cid:35) (261)The normalization condition for the density ρ ( R ) yields the following condition for any order m ≥
10 = (cid:90) + ∞ dRρ m ( R ) (262)This condition determines the constant C m , and the solution of Eq. 261 can be then rewritten with the notation Z de Eq. 231 ρ m ( R ) = e − U ( R ) Z (cid:90) + ∞ dR (cid:48)(cid:48) e − U ( R (cid:48)(cid:48) ) (cid:90) RR (cid:48)(cid:48) dR (cid:48) e U ( R (cid:48) ) Υ m ( R (cid:48) ) (263)i.e. more explicitly for m = 1 , , , m ( R ) ρ ( R ) = e − U ( R ) Z (cid:90) + ∞ dR (cid:48)(cid:48) e − U ( R (cid:48)(cid:48) ) (cid:90) RR (cid:48)(cid:48) dR (cid:48) e U ( R (cid:48) ) (cid:20) G ( R (cid:48) ) ρ eqst ( R (cid:48) ) D ( R (cid:48) ) (cid:21) (264) ρ ( R ) = e − U ( R ) Z (cid:90) + ∞ dR (cid:48)(cid:48) e − U ( R (cid:48)(cid:48) ) (cid:90) RR (cid:48)(cid:48) dR (cid:48) e U ( R (cid:48) ) (cid:20) G ( R (cid:48) ) ρ eqst ( R (cid:48) ) + G ( R (cid:48) ) ρ ( R (cid:48) ) D ( R (cid:48) ) (cid:21) ρ ( R ) = e − U ( R ) Z (cid:90) + ∞ dR (cid:48)(cid:48) e − U ( R (cid:48)(cid:48) ) (cid:90) RR (cid:48)(cid:48) dR (cid:48) e U ( R (cid:48) ) (cid:20) G ( R (cid:48) ) ρ eqst ( R (cid:48) ) + G ( R (cid:48) ) ρ ( R (cid:48) ) + G ( R (cid:48) ) ρ ( R (cid:48) ) D ( R (cid:48) ) (cid:21) ρ ( R ) = e − U ( R ) Z (cid:90) + ∞ dR (cid:48)(cid:48) e − U ( R (cid:48)(cid:48) ) (cid:90) RR (cid:48)(cid:48) dR (cid:48) e U ( R (cid:48) ) (cid:20) G ( R (cid:48) ) ρ eqst ( R (cid:48) ) + G ( R (cid:48) ) ρ ( R (cid:48) ) + G ( R (cid:48) ) ρ ( R (cid:48) ) + G ( R (cid:48) ) ρ ( R (cid:48) ) D ( R (cid:48) ) (cid:21) VIII. CONCLUSION
In this paper, we have revisited the large deviations properties of the finite-time Lyapunov exponent for the 2D ma-trix Langevin dynamics in relation with the recent progresses made in the field of large deviations for non-equilibriumstochastic processes, where additive functionals can be analyzed from two points of view. In the first approach, onestarts from the large deviations at level 2.5 for the joint probability of the empirical density and of the empirical currentof the Riccati process in order to compute the cumulant generating function of the Lyapunov exponent via some Euler-Lagrange optimization. We have described in detail how this optimization procedure can be solved perturbativelyin order to obtain explicitly the first cumulants, both when the Riccati steady state is a non-equilibrium state withcurrent or an equilibrium state without current. We have then discussed the second approach, where the cumulantgenerating function is obtained via the spectral analysis of the appropriate tilted Fokker-Planck operator. We haveexplained how the associated conditioned process constructed via the generalization of Doob’s h-transform is usefulto clarify the equivalence with the first approach. Finally, we have applied this general framework to one-dimensionalAnderson Localization models with random scalar potential and with random supersymmetric potential.
Acknowledgements
It is a pleasure to thank Christophe Texier for his explanations [124] on the relations between the numerous resultsobtained in his papers [15, 16, 18–20] concerning the large deviations properties of Lyapunov exponents in variousmodels.6
Appendix A: Explicit first cumulants for the case of a Riccati non-equilibrium steady-state
In this Appendix, we consider the case where the potential U ( R ) introduced in Eq. 48 is non-periodic on the Riccatiring (see Eq. 73) and where the steady state ρ st is a non-equilibrium steady state ρ neqst with a steady current j st (seesubsection III E). We describe how the optimization procedure in two steps of subsection IV F can be implemented atthe level of the perturbation theory in the parameter k .
1. Perturbative solution in k for the effective force G ( R ) and the scaled cumulant generating function µ ( k ) Plugging the perturbative expansions for the effective force G ( R ) and for the scaled cumulant generating function µ ( k ) G ( R ) = F ( R ) + kG ( R ) + k G ( R ) + k G ( R ) + k G ( R ) + O ( k ) µ ( k ) = kµ + k µ + k µ + k µ + O ( k ) (A1)into the Euler-Lagrange Eq. 102 yields the following differential equations order by order in terms of the notation U (cid:48) ( R ) = − F ( R ) D ( R ) of Eq 48 G (cid:48) ( R ) − U (cid:48) ( R ) G ( R ) = 2 µ − α ( R ) ≡ Ω ( R ) G (cid:48) ( R ) − U (cid:48) ( R ) G ( R ) = 2 µ − G ( R )2 D ( R ) ≡ Ω ( R ) G (cid:48) ( R ) − U (cid:48) ( R ) G ( R ) = 2 µ − G ( R ) G ( R ) D ( R ) ≡ Ω ( R ) G (cid:48) ( R ) − U (cid:48) ( R ) G ( R ) = 2 µ − G ( R ) G ( R ) D ( R ) − G ( R )2 D ( R ) ≡ Ω ( R ) (A2)while the condition of Eq. 103 gives β tot = (cid:90) + ∞−∞ dR G ( R )2 D ( R )0 = (cid:90) + ∞−∞ dR G m ( R )2 D ( R ) for m ≥ m ≥
1, the solution G m ( R ) for the effective force at order k m should be periodic on the infinite Riccatiring. Since we wish to write the generic solution for any case, let us first regularize the problem on a finite ring R ∈ [ R min , R max ] as in Eq. 75 to write the solution for G m ( R ) as a function of the inhomogeneous term Ω m ( R ) ofEq. A2 as G m ( R ) = e U ( R ) (cid:104) e U ( R min ) (cid:82) RR min dR (cid:48) Ω m ( R (cid:48) ) e − U ( R (cid:48) ) + e U ( R max ) (cid:82) R max R dR (cid:48) Ω m ( R (cid:48) ) e − U ( R (cid:48) ) (cid:105)(cid:2) e U ( R min ) − e U ( R max ) (cid:3) (A4)The conditions of Eq. A3 then involve integrals of the form (cid:90) R max R min dR G m ( R )2 D ( R ) = (cid:82) R max R min dR e U ( R ) D ( R ) (cid:104) e U ( R min ) (cid:82) RR min dR (cid:48) Ω m ( R (cid:48) ) e − U ( R (cid:48) ) + e U ( R max ) (cid:82) R max R dR (cid:48) Ω m ( R (cid:48) ) e − U ( R (cid:48) ) (cid:105)(cid:2) e U ( R min ) − e U ( R max ) (cid:3) = (cid:82) R max R min dR (cid:48) Ω m ( R (cid:48) )2 e − U ( R (cid:48) ) (cid:104) e U ( R min ) (cid:82) R max R (cid:48) dR e U ( R ) D ( R ) + e U ( R max ) (cid:82) R (cid:48) R min dR e U ( R ) D ( R ) (cid:105)(cid:2) e U ( R min ) − e U ( R max ) (cid:3) = (cid:82) R max R min dR (cid:48) Ω m ( R (cid:48) )2 e − U ( R (cid:48) ) (cid:104) e − U ( R max ) (cid:82) R max R (cid:48) dR e U ( R ) D ( R ) + e − U ( R min ) (cid:82) R (cid:48) R min dR e U ( R ) D ( R ) (cid:105)(cid:2) e − U ( R max ) − e U ( R min ) (cid:3) = (cid:90) R max R min dR (cid:48) Ω m ( R (cid:48) )2 (cid:18) ρ neqst ( R (cid:48) ) j st (cid:19) (A5)7where we have recognized the steady state solution ρ neqst ( R ) of Eq. 75 and the corresponding steady state current j st .So the coefficients µ m of the expansion of Eq. A1 for the scaled cumulant generating function µ ( k ) that appear inthe inhomogeneous terms Ω m ( R ) of Eq. A2 are determined order by order by Eq. A3 using Eq A5 β tot = (cid:90) R max R min dR G ( R )2 D ( R ) = (cid:90) R max R min dR Ω ( R )2 (cid:18) ρ neqst ( R ) j st (cid:19) = (cid:90) R max R min dR [ µ − α ( R )] (cid:18) ρ neqst ( R ) j st (cid:19) (A6)0 = (cid:90) R max R min dR G ( R )2 D ( R ) = (cid:90) R max R min dR Ω ( R )2 (cid:18) ρ neqst ( R ) j st (cid:19) = (cid:90) R max R min dR (cid:20) µ − G ( R )4 D ( R ) (cid:21) (cid:18) ρ neqst ( R ) j st (cid:19) (cid:90) R max R min dR G ( R )2 D ( R ) = (cid:90) R max R min dR Ω ( R )2 (cid:18) ρ neqst ( R ) j st (cid:19) = (cid:90) R max R min dR (cid:20) µ − G ( R ) G ( R )2 D ( R ) (cid:21) (cid:18) ρ neqst ( R ) j st (cid:19) (cid:90) R max R min dR G ( R )2 D ( R ) = (cid:90) R max R min dR Ω ( R )2 (cid:18) ρ neqst ( R ) j st (cid:19) = (cid:90) R max R min dR (cid:20) µ − G ( R ) G ( R )2 D ( R ) − G ( R )4 D ( R ) (cid:21) (cid:18) ρ neqst ( R ) j st (cid:19) The normalization of the steady state solution ρ neqst ( R ) on [ R min , R max ] allows to rewrite these equations as µ = (cid:90) R max R min dRα ( R ) ρ neqst ( R ) + β tot j st = λ typ µ = (cid:90) R max R min dR (cid:20) G ( R )4 D ( R ) (cid:21) ρ neqst ( R ) µ = (cid:90) R max R min dR (cid:20) G ( R ) G ( R )2 D ( R ) (cid:21) ρ neqst ( R ) µ = (cid:90) R max R min dR (cid:20) G ( R ) G ( R )2 D ( R ) − G ( R )4 D ( R ) (cid:21) ρ neqst ( R ) (A7)The first equation corresponds to the typical value λ typ of Eq. 85 as it should (Eq 38). The other equations allow tocompute µ m =2 , , in terms of the solutions G m ( R ) given in Eq A4.In the limit R min → −∞ and R max → + ∞ , one then needs to take into account the behavior of the potential U ( R ) for R → ±∞ to obtain the appropriate solution on the infinite Riccati ring. One example is described below insubsection A 3.
2. Perturbative solution in k for the optimal density ρ ( R ) and the optimal current j Even if the perturbative solution for the scaled cumulant generating function µ ( k ) has already been obtained inEq. A7, it is nevertheless interesting to derive the corresponding perturbative solutions in k for the optimal density ρ ( R ) and the optimal current jρ ( R ) = ρ neqst ( R ) + kρ ( R ) + k ρ ( R ) + k ρ ( R ) + k ρ ( R ) + O ( k ) j = j st + kj + k j + k j + k j + O ( k ) (A8)Plugging these perturbative expansions and the perturbative expansion for the optimal force G ( R ) (Eq. A1) into Eq.101 with the notation U (cid:48) ( R ) = − F ( R ) D ( R ) of Eq 48 leads to the following differential equations order by order ρ (cid:48) ( R ) + U (cid:48) ( R ) ρ ( R ) = G ( R ) ρ neqst ( R ) − j D ( R ) ≡ Υ ( R ) ρ (cid:48) ( R ) + U (cid:48) ( R ) ρ ( R ) = G ( R ) ρ neqst ( R ) + G ( R ) ρ ( R ) − j D ( R ) ≡ Υ ( R ) ρ (cid:48) ( R ) + U (cid:48) ( R ) ρ ( R ) = G ( R ) ρ neqst ( R ) + G ( R ) ρ ( R ) + G ( R ) ρ ( R ) − j D ( R ) ≡ Υ ( R ) ρ (cid:48) ( R ) + U (cid:48) ( R ) ρ ( R ) = G ( R ) ρ neqst ( R ) + G ( R ) ρ ( R ) + G ( R ) ρ ( R ) + G ( R ) ρ ( R ) − j D ( R ) ≡ Υ ( R ) (A9)8For any m ≥
1, the solution ρ m ( R ) for the density at order k m should be periodic on the Riccati ring, so one canwrite the solution in terms of the inhomogeneous terms Υ m ( R ) of Eq. A9 in the form analogous to Eq. 75 ρ neqm ( R ) = e − U ( R ) (cid:104) e − U ( R min ) (cid:82) RR min dR (cid:48) Υ m ( R (cid:48) ) e U ( R (cid:48) ) + e − U ( R max ) (cid:82) R max R dR (cid:48) Υ m ( R (cid:48) ) e U ( R (cid:48) ) (cid:105)(cid:2) e − U ( R min ) − e − U ( R max ) (cid:3) (A10)The normalization condition for the density ρ ( R ) yields the following condition for any order m ≥
10 = (cid:90) R max R min dRρ m ( R ) = (cid:82) R max R min dRe − U ( R ) (cid:104) e − U ( R min ) (cid:82) RR min dR (cid:48) Υ m ( R (cid:48) ) e U ( R (cid:48) ) + e − U ( R max ) (cid:82) R max R dR (cid:48) Υ m ( R (cid:48) ) e U ( R (cid:48) ) (cid:105)(cid:2) e − U ( R min ) − e − U ( R max ) (cid:3) = (cid:82) R max R min dR (cid:48) Υ m ( R (cid:48) ) e U ( R (cid:48) ) (cid:104) e − U ( R min ) (cid:82) R max R (cid:48) dRe − U ( R ) + e − U ( R max ) (cid:82) R (cid:48) R min dRe − U ( R ) (cid:105)(cid:2) e − U ( R min ) − e − U ( R max ) (cid:3) = (cid:90) R max R min dR (cid:48) Υ m ( R (cid:48) ) e U ( R (cid:48) ) e U ( R max ) (cid:82) R max R (cid:48) dRe − U ( R ) + e U ( R min ) (cid:82) R (cid:48) R min dRe − U ( R ) e U ( R max ) − e U ( R min ) (A11)These equations determine order by order the current coefficients j m that appear in the inhomogeneous terms Υ m ( R )of Eq. A9. It is thus convenient to introduce the probability distributionΠ( R (cid:48) ) ≡ ( − j st ) e U ( R (cid:48) ) D ( R (cid:48) ) e U ( R max ) (cid:82) R max R (cid:48) dRe − U ( R ) + e U ( R min ) (cid:82) R (cid:48) R min dRe − U ( R ) e U ( R max ) − e U ( R min ) (A12)that is normalized as a consequence of Eq. 76 defining the stationary current j st (cid:90) R max R min dR (cid:48) Π( R (cid:48) ) = ( − j st ) e U ( R max ) (cid:82) R max R min dR (cid:48) e U ( R (cid:48) ) D ( R (cid:48) ) (cid:82) R max R (cid:48) dRe − U ( R ) + e U ( R min ) (cid:82) R max R min dR (cid:48) e U ( R (cid:48) ) D ( R (cid:48) ) (cid:82) R (cid:48) R min dRe − U ( R ) e U ( R max ) − e U ( R min ) = ( − j st ) e U ( R max ) (cid:82) R max R min dRe − U ( R ) (cid:82) RR min dR (cid:48) e U ( R (cid:48) ) D ( R (cid:48) ) + e U ( R min ) (cid:82) R max R min dRe − U ( R ) (cid:82) R max R dR (cid:48) e U ( R (cid:48) ) D ( R (cid:48) ) e U ( R max ) − e U ( R min ) = ( − j st ) (cid:104) e − U ( R min ) (cid:82) R max R min dRe − U ( R ) (cid:82) RR min dR (cid:48) D ( R (cid:48) ) e U ( R (cid:48) ) + e − U ( R max ) (cid:82) R max R min dRe − U ( R ) (cid:82) R max R dR (cid:48) D ( R (cid:48) ) e U ( R (cid:48) ) (cid:105)(cid:2) e − U ( R min ) − e − U ( R max ) (cid:3) = 1 (A13)The solutions of Eqs A11 for the current coefficients j m can be then rewritten as j = (cid:90) R max R min dR [ G ( R ) ρ neqst ( R )] Π( R ) j = (cid:90) R max R min dR [ G ( R ) ρ neqst ( R ) + G ( R ) ρ ( R )] Π( R ) j = (cid:90) R max R min dR (cid:48) [ G ( R ) ρ neqst ( R ) + G ( R ) ρ ( R ) + G ( R ) ρ ( R ) − ] Π( R ) j = (cid:90) R max R min dR [ G ( R ) ρ neqst ( R ) + G ( R ) ρ ( R ) + G ( R ) ρ ( R ) + G ( R ) ρ ( R )] Π( R ) (A14)In the limit R min → −∞ and R max → + ∞ , one then needs to take into account the behavior of the potential U ( R )for R → ±∞ to obtain the appropriate solution on the infinite Riccati ring. Let us now describe one example.
3. Perturbative solution when the potential difference diverges U (+ ∞ ) − U ( −∞ ) = + ∞ When the potential difference diverges U (+ ∞ ) − U ( −∞ ) = + ∞ , we have described the non-equilibrium steadystate in the subsection III E 1, and it is thus interesting to write the corresponding perturbative solution in k .9In the solution of Eq. A4, only the right terms survive in the numerator and denominator in the limit R min → −∞ and R max → + ∞ G m ( R ) = − e U ( R ) (cid:90) + ∞ R dR (cid:48) Ω m ( R (cid:48) ) e − U ( R (cid:48) ) (A15)i.e. more explicitly for m = 1 , , , G ( R ) = − e U ( R ) (cid:90) + ∞ R dR (cid:48) [2 µ − R (cid:48) ] e − U ( R (cid:48) ) G ( R ) = − e U ( R ) (cid:90) + ∞ R dR (cid:48) (cid:20) µ − G ( R (cid:48) )2 D (cid:21) e − U ( R (cid:48) ) G ( R ) = − e U ( R ) (cid:90) + ∞ R dR (cid:48) (cid:20) µ − G ( R (cid:48) ) G ( R (cid:48) ) D (cid:21) e − U ( R (cid:48) ) G ( R ) = − e U ( R ) (cid:90) + ∞ R dR (cid:48) (cid:20) µ − G ( R (cid:48) ) G ( R (cid:48) ) D − G ( R (cid:48) )2 D (cid:21) e − U ( R (cid:48) ) (A16)while Eq A7 read µ = (cid:90) + ∞−∞ dRRρ neqst ( R ) = λ typ µ = (cid:90) + ∞−∞ dR (cid:20) G ( R )4 D (cid:21) ρ neqst ( R ) µ = (cid:90) + ∞−∞ dR (cid:20) G ( R ) G ( R )2 D (cid:21) ρ neqst ( R ) µ = (cid:90) + ∞−∞ dR (cid:20) G ( R ) G ( R )2 D − G ( R )4 D (cid:21) ρ neqst ( R ) (A17)So the iterative procedure goes as follows : one plugs µ = λ typ into A4 to compute G ( R ) that can be then pluggedinto Eq A17 to compute µ , that can be then plugged into Eq A4 to compute G ( R ), and so on.In the solution of Eq. A10, only the left terms survive in the numerator and denominator in the limit R min → −∞ and R max → + ∞ ρ m ( R ) = e − U ( R ) (cid:90) R −∞ dR (cid:48) Υ m ( R (cid:48) ) e U ( R (cid:48) ) (A18)i.e. more explicitly for m = 1 , , , ρ ( R ) = e − U ( R ) (cid:90) R −∞ dR (cid:48) (cid:20) G ( R ) ρ neqst ( R ) − j D (cid:21) e U ( R (cid:48) ) ρ ( R ) = e − U ( R ) (cid:90) R −∞ dR (cid:48) (cid:20) G ( R ) ρ neqst ( R ) + G ( R ) ρ ( R ) − j D (cid:21) e U ( R (cid:48) ) ρ ( R ) = e − U ( R ) (cid:90) R −∞ dR (cid:48) (cid:20) G ( R ) ρ neqst ( R ) + G ( R ) ρ ( R ) + G ( R ) ρ ( R ) − j D (cid:21) e U ( R (cid:48) ) ρ ( R ) = e − U ( R ) (cid:90) R −∞ dR (cid:48) (cid:20) G ( R ) ρ neqst ( R ) + G ( R ) ρ ( R ) + G ( R ) ρ ( R ) + G ( R ) ρ ( R ) − j D (cid:21) e U ( R (cid:48) ) (A19)In terms of the function of Eq. A12 Π( R (cid:48) ) ≡ ( − j st ) e U ( R (cid:48) ) D ( R (cid:48) ) (cid:90) + ∞ R (cid:48) dRe − U ( R ) (A20)0the current coefficients read j = (cid:90) + ∞−∞ dR [ G ( R ) ρ neqst ( R )] Π( R ) j = (cid:90) + ∞−∞ dR [ G ( R ) ρ neqst ( R ) + G ( R ) ρ ( R )] Π( R ) j = (cid:90) + ∞−∞ dR (cid:48) [ G ( R ) ρ neqst ( R ) + G ( R ) ρ ( R ) + G ( R ) ρ ( R ) − ] Π( R ) j = (cid:90) + ∞−∞ dR [ G ( R ) ρ neqst ( R ) + G ( R ) ρ ( R ) + G ( R ) ρ ( R ) + G ( R ) ρ ( R )] Π( R ) (A21)So here the iterative procedure goes as follows : one first compute j to plug it into into Eq. A19 to obtain ρ ( R )that can be then plugged into Eq. A21 to compute j , that can be then plugged into Eq A19 to compute ρ ( R ), andso on. Appendix B: Explicit first cumulants for the case of a Riccati equilibrium steady-state