The information geometry of 2-field functional integrals

1. [1]

   of the functional integral (46), which satasify (∑ i ∂ ˙φ† i ∂φ† i + ∑ i ∂ ˙φi ∂φi )⏐ ⏐ ⏐ ⏐ ⏐ ¯φ† , ¯φ = 0 (61) Eq. (

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2. [2]

   wt is a density of rays for joint base distributions and likelihood ratios that is conserved along the Doi-Peliti stationary trajectories

   may thus be recast as the conservation law for a 2D-dimensional current ( ˙φ†w, ˙φw ) , 0 = ∂wt ∂t + ∑ i ∂ ∂φ† i ( ˙φ† i wt ) + ∑ i ∂ ∂φi ( ˙φiwt ) (62) which is Liouville’s theorem. wt is a density of rays for joint base distributions and likelihood ratios that is conserved along the Doi-Peliti stationary trajectories. log wt is the leading exponential a...

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3. [3]

   of the Wigner function in terms of the functional integral is convenient to manipu- late but perhaps not very self-explanatory. App. D gives a direct construction of the stationary-point approxima- tion in terms of a density ρ(θ) over the basis of coherent states|φ) and their Laplace transforms, and verifies that the sequence of stationary points do indeed...

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4. [4]

   Then the transport equation (

   is defined in the species basis, it will not be possible to factor out non-dynamical combinations. Then the transport equation (

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5. [5]

   for the current of the Wigner density will occupy only a sub-manifold of the 2D-dimensional Doi-Peliti coordinate space needed to de- fine the system. A convenient way to handle constraints is to work in the eigenbasis of the Fisher metric which we will index with subscript α, where an action-angle counterpart to the transport equation (

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6. [6]

   reads 0 = ∂wt ∂t + ∑ α ∂ ∂θ α ( ˙θαwt ) + ∑ α ∂ ∂nα ( ˙nαwt) (63) The picture of the Liouville equation as implying a con- served volume element d dt (∏ α δθαδnα ) = 0 (64) with the product index α taken only over nonzero eigen- values of the Fisher metric, remains nondegenerate and has a direct interpretation in terms of the product of eigenvectors of th...

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7. [7]

   suggests that the 2D-dimensional differential of the stationary-point CGF should likewise obey a symplectic transport law, imply- ing a transport law for the Fisher metric. To derive those results we return to the expression of the differen- tial of the CGF in terms of the generalized Pythagorean 13 theorem ( 16), and derive the Fisher metric from the ψ- di...

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8. [8]

   and ( 53). To study their independent variations about a reference value (θR, ηR), introduce two distinct exponential families, labeled ρ≡ ˜ρ(θ, η)η=ηR ρ′≡ ˜ρ(θ, η)θ=θR (65) The ψ-divergence Dψ(θ : η) = ψ(θ) + ψ∗ (n)− nθ, a Bregman divergence of the CGF, is related to the Kullback-Leibler divergence of ρ′ from ρ as Dψ(θ : η) = DKL(ρ′∥ ρ) = ∑ n ˜ρn(θR, η) ...

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9. [9]

   Two third-order mixed partials define the connec- tion coefficients for Amari’s dually-flat connections on exponential and mixture coordinates

   in dual exponential coordinates. Two third-order mixed partials define the connec- tion coefficients for Amari’s dually-flat connections on exponential and mixture coordinates. Written in all- contravariant indices,8 these are given by ΓD kij = ∂ ∂θ k gD ij =− ∂2ψ ∂θ k∂θ i ∑ n ˜ρn(θR, η) ( nj− ∂ψ ∂η j ) ΓD∗ kij = ∂ ∂η k gD ij = ∑ n ˜ρn(θR, η) ( nj− ∂ψ ∂η j )2...

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10. [10]

    Introduce two vector fields corresponding to variations in θ at the final time T , and to variations in η at the initial time 0

    for the Wigner function. Introduce two vector fields corresponding to variations in θ at the final time T , and to variations in η at the initial time 0. The first can be independently imposed 8 Note that it is the dual connection ΓD∗ kij ≡ 0, written in all- covariant indices, which vanishes as the affine connection on the mixture family. 14 through the argum...

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11. [11]

    and its equivalence to the Hessian definition of g in Eq. ( 69), (δθ)igij(δη)j≡ (δθ)i(δηn)i ≡ (δθn)j(δη)j (74) Although the field variable n is the same in either action- angle transform ( 48) or ( 53), the two displacements δηn and δθn are independent vector fields

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12. [12]

    ( 73) has a symmetric form but evolves δθ and δη respectively using L and ˜L, making it not immediately apparent that the inner product is preserved

    The conserved inner product of dual vector fields, and directional transport of the metric Eq. ( 73) has a symmetric form but evolves δθ and δη respectively using L and ˜L, making it not immediately apparent that the inner product is preserved. Writing the field δη in its dual mixture coordinate as in the first line of Eq. (

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13. [13]

    the time derivative becomes ( d dt δn ) i = δn0j ∂ ∂n0j ˙ni(θT , n0, t) = δntj ∂ ∂ntj ˙ni(θT , n0, t) =− δntj ∂2L ∂nj∂θ i (75) The condition ( 22) is met and we have ( d dt δθ )j (δn)j + (δθ)i ( d dt δn ) i = 0 (76) Using Eq. ( 76) to evaluate the change in the inner product written as (δθ)igij(δη)j, substituting the deriva- tives (50) for ˙θ and ˙η, and ...

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14. [14]

    Conservation of the inner product through the combined effects of two maps The inner product (

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15. [15]

    is preserved through the complementary action of two maps, one generated by the time-dependence of θ, and the other by the time- dependence of η. By construction, δθ depends on time only through ˙θ, and δη only through ˙η, while the met- ric has no explicit time dependence but changes under both maps as the location (θ, η) changes. Denoting by d/dt|˙θ and...

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16. [16]

    The dual connections (

    Referencing arbitrary dual connections to dually flat connections in the exponential family The manifold for a Doi-Peliti system with D indepen- dent components has dimension 2D, with parallel sub- spaces for the base distribution and likelihood ratio. The dual connections (

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17. [17]

    ( 68), which act within the same D- dimensional exponential family

    act within these two independent subspaces, in contrast to the dually-flat connections ΓD and ΓD∗ of Eq. ( 68), which act within the same D- dimensional exponential family. Although the Fisher metric is a function only of the overall importance dis- tribution, which aggregates dependence from the base distribution and likelihood, the symplectic transforma-...

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18. [18]

    ( 83) gives expressions for the dual covariant derivatives of the Fisher metric ∇ (θ) k gij = Tkji− Γ(θ) kij ∇ (η)∗ k gij = Tkji− Γ(η)∗ kji (85)

    into Eq. ( 83) gives expressions for the dual covariant derivatives of the Fisher metric ∇ (θ) k gij = Tkji− Γ(θ) kij ∇ (η)∗ k gij = Tkji− Γ(η)∗ kji (85)

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19. [19]

    Flat connections for coherent-state coordinates Of particular interest in Doi-Peliti theory will be the canonical transformations ( 48) and ( 53) between coherent-state and number-potential (or action-angle) coordinates. We note that the forms of the connection coefficients for which affine transport in fields φ† is flat in the likelihood subspace, and affine tra...

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20. [20]

    different from both the Levi-civita connection and the dually-flat con- nections (

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21. [21]

    of Nagaoka and Amari [13, 14]. A. T wo-argument and one-argument generating functions on distributions with a conserved quantity The two-state model describes (for example) a one- particle chemical reaction in a well-mixed reactor with the schema a k+ ⇋ k− b (89) The probability per unit time for a reaction event is given by rate constants k+ and k− , and...

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22. [22]

    Here for simplicity we will take k+ and k− to be fixed

    will retain that form at all times under the master equation for the schema ( 89), even with time-dependent coefficients. Here for simplicity we will take k+ and k− to be fixed. There- for the distribution at any time is specified by descaled mean values νa =⟨na⟩ /N, νb =⟨nb⟩ /N, with νa+νb = 1. Although the system has only one dynamical degree of freedom, it...

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23. [23]

    for the binomial distribution is eψ(log z)≡ ∑ n zna a znb b ρna,nb = [zaνa + zbνb]N (90) Because the total number N = (n b + na) is fixed, the normalized 1-variable distribution may be written ρn =√ νbνa N ( N n ) (νb νa )n (91) and the terms in the generating function ( 90) regrouped as eψ(log z) =√ zbza N ∑ n ρn ( zb za )n (92) Introducing rotated coordi...

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24. [24]

    after conversion to field variables, which is L = k+ ( φ† a− φ† b ) φa + k− ( φ† b− φ† a ) φb (95) In what follows, math boldface will be reserved for pa- rameters in the generator such as k± or functions of these such as the associated steady states used in Eq. ( 51). Two descalings reduce the problem to parameters which are dimensionless ratios. The first...

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25. [25]

    18 Conservation of total number N results in a gener- atorL that is a function only of the difference variable( φ† b− φ† a )

    in terms of relative hopping rates, k+ k+ + k− = nb N ≡ νb k− k+ + k− = na N ≡ νa (97) As for the discrete index n, define ν≡ (νb− νa) /2. 18 Conservation of total number N results in a gener- atorL that is a function only of the difference variable( φ† b− φ† a ) . Therefore it is natural to rotate the coherent- state fields to components corresponding to co...

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26. [26]

    Absence of the field Φ† from ˆL implies constancy of the expectation for ˆΦ.10

    the action ( 45) becomes S = N ∫ dτ [ − ∂τ Φ† ˆΦ− ∂τ φ† ˆφ + φ† ( ˆφ− ν ˆΦ )] ≡ N ∫ dτ ( − ∂τ Φ† ˆΦ− ∂τ φ† ˆφ + ˆL ) (99) A descaled Liouville function has been introduced as N (k+ + k− ) ˆL ≡ L. Absence of the field Φ† from ˆL implies constancy of the expectation for ˆΦ.10

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27. [27]

    Therefore the coherent- state variables cannot directly be interpreted as mean values of number variables in the nominal distribution or mean weights in its dual likelihood ratio

    Splitting the symplectic structure between coherent-st ate conjugate field pairs Although ˆΦ obeys certain time-translation invariances in correlation functions, its value even along stationary paths will not generally be 1. Therefore the coherent- state variables cannot directly be interpreted as mean values of number variables in the nominal distribution...

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28. [28]

    Recall that the instantaneous steady state under the generating process is the scale variable for the dualiz- ing canonical transform ( 51)

    are given by 1 N φ† bφb≡ νb≡ (1 2 + ν ) 1 N φ† aφa≡ νa≡ (1 2− ν ) (100) 10 It implies constancy of a tower of higher-order correlation func- tions expressing exact conservation of the underlying vari able N , though we do not develop the 2FFI representation of correlat ion functions in this paper. Recall that the instantaneous steady state under the gener...

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29. [29]

    as 1 2 ( φ† bφb− φ† aφa ) ( φ† bφb + φ† aφa )≡ (νb− νa) 2 ≡ ν (101) The action of the tilt alone can be isolated, without regard to the underlying nominal distribution, by ref- erencing the action of the φ† fields to the steady state rather than to φ, defining an offset ν as 1 2 ( φ† bνb− φ† aνa ) ( φ† bνb + φ† aνa )≡ (ν b− νa) 2 ≡ ν (102) Likewise, the mean...

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30. [30]

    Stationary-path solutions and Liouville volume element Solutions to the stationary-path equations of mo- tion ( 47) for the Liouville function ( 95) are evaluated in App. E 1. Stationary-path approximations to the time-dependent density ρ would be binomial distributions even if the ex- act ρ were not (the stationary point is always a pure co- herent state...

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31. [31]

    carries the mean value ¯ν0 in the starting density ρ0, imposed as an initial-data parameter. It is through this function that the final-time tilt data in the form of the parameter ν T , propagated forward to the stationary-path values of φ† a0 and φ† b0, determines the stationary path values for the fields φ of the base distribution, establishing the poten-...

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32. [32]

    Invariant cumulant-generating function and the incompressible phase-space density The stationary value of ˆΦ0, obtained from the gradi- ent of ψ0 with respect to the components φ† 0a and φ† 0b, is computed in Eq. ( E8). It differs from unity – the rea- son constructions ( 102) and ( 103) were needed – and it is equal to the stationary value of ˆΦ at all ti...

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33. [33]

    ( 106) is the basis for all information densities in this simple linear system

    and evaluates to the constant ψ N = log [ ν a νa ¯νa + νb νb ¯νb ] =− log ˆΦ0 (107) ˆΦ0 in Eq. ( 106) is the basis for all information densities in this simple linear system. Through the stationary- point relation (

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34. [34]

    between the Wigner function and the CGF,− log ˆΦ0 is the incompressible phase-space density convected along stationary trajectories by Eq. ( 62). As shown below, it is also the geometrically invariant part of sole nonzero eigenvalue of the Fisher metric. C. Fisher metric The Fisher metric (

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35. [35]

    for the 2-state system evaluates, along the stationary path at any time, to g N = ∂ν ∂θ [ 1 − 1 ] [ 1 − 1 ] (108) The nonzero eigenvalue comes from the single-argument generating function in Eq. (

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36. [36]

    The term ∂ν/∂θ in Eq

    for the difference coor- dinate n, and the zero eigenvalue comes from the linear CGF hN for the conserved quantity N . The term ∂ν/∂θ in Eq. ( 108) may be converted, after some algebra, to the form ∂ν ∂θ = (1 4− ν2) (1 4− ¯ν2) (1 4− ν 2) ˆΦ2 0 (109) The measure terms ( 1/4− ν 2) and ( 1/4− ¯ν2) appearing in Eq. ( 109) follow the divisions ( 102) and ( 103)...

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37. [37]

    V B from the ψ-divergence and to dually- symplectic parallel transport, we first express the base and tilt displacements (

    to the construc- tion of Sec. V B from the ψ-divergence and to dually- symplectic parallel transport, we first express the base and tilt displacements (

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38. [38]

    20 Introduce reference values for the fields θ and h defined in Eq

    and ( 103) in terms of the coordinates in their respective exponential families. 20 Introduce reference values for the fields θ and h defined in Eq. ( 93), corresponding to the steady-state measure under the parameters of the generating process, denoted θ≡ log ( 1 2 + ν 1 2− ν ) h≡ 1 2 log (1 4− ν 2 ) =− log [ 2 ch θ 2 ] (110) It is clear, in the 2-argument...

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39. [39]

    and not the Fisher geometry captures the special role of coherent states. A divergence under the Hessian of ψ in coherent-state variables, which we will denote ∆δs2 for reasons to be- come clear in a moment, if converted from the coordi- nates ν to coordinates θ along the z-affine contour ( 112), evaluates as 1 N ∆ds2≡ (δθ)2 (1 4− ν2 )2 ∂2 ∂ν 2 ( ψ N ) ≡− (...

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40. [40]

    Connection coefficients and absorption of measure terms In this linear model, time evolution of φ† and φ has no cross-dependence once the initial values have been fixed through the gradients of ψ0 as explained in Sec. VI B 2. Thus ( ∇ (η) k δθ )j = 0 and ( ∇ (θ)∗ k δη )j = 0. App. E 4 computes connection coefficients and covari- ant derivatives for the vector ...

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41. [41]

    The covariant part of the change in the Fisher metric, from Eq

    of the coherent-state fields in the exponential-family coordi- nates. The covariant part of the change in the Fisher metric, from Eq. (

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42. [42]

    ( E23) to be ˙θ∇ θg = ( ˙v ∂ ∂v log ˆΦ2 0 ) g ˙η∇ ∗ ηg = ( ˙u ∂ ∂u log ˆΦ2 0 ) g (124) Only the dependence in the Fisher eigenvalue ˆΦ2 0 from Eq

    is computed in Eq. ( E23) to be ˙θ∇ θg = ( ˙v ∂ ∂v log ˆΦ2 0 ) g ˙η∇ ∗ ηg = ( ˙u ∂ ∂u log ˆΦ2 0 ) g (124) Only the dependence in the Fisher eigenvalue ˆΦ2 0 from Eq. (

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43. [43]

    The two lines of Eq

    appears. The two lines of Eq. ( 124) (which are equal and oppo- site) scale as ∼ e− T , and have an interpretation similar to that of a Le Chatelier principle. The term Λe− T in Eq. (106) for ˆΦ0 is a susceptibility of the initial stationary value φ0 to the perturbation by the tilt variable φ† T = z, attenuated exponentially from time T to time 0. The rol...

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44. [44]

    The two con- tributions group as 0 = d dt (δθ g δη ) = ( ∂ ∂t δθ ) δην + ˙θ∇ (θ) θ (δθ g) δη + δθν ( ∂ ∂t δη ) + δθ ˙η∇ (η)∗ η (g δη) (125) 22 The two rows of Eq

    Duality of dynamics and inference in Doi-Peliti theory The natural separation of the coordinate transforma- tion of the inner product of vector fields δθ and δη gener- ated by time translation is not between exponential and mixture coordinates, as in the dually-flat connections of Amari [11], but rather between the symplectically dual contributions from cha...

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45. [45]

    captures in the clearest way possible the sym- plectic balance of distribution dynamics (through η) and inference (through θ) in Doi-Peliti theory, through both the direct effects of the exponential growth and decay eigenvalues (± 1) and the Le Chatelier-like susceptibility of the density ˆΦ0. G. The Fisher information density and large-deviation ratios as...

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46. [46]

    that the probability for the value n of a sample to exceed a threshold n is given in terms of the large-deviation function by P (n≥ n|η)∼ e− ψ∗ (n;η) (127) In a 1-dimensional system, 11 for two threshold values nB > n A, the conditional probability for n to surpass nB given that it has surpassed nA is the ratio P (nB|nA; η)≡ P (n≥ nB|η) P (n≥ nA|η) ∼ e− [...

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47. [47]

    may be computed at any time, for instance the final time T when the thresholds nB and nA are imposed as experimental conditions, and η2 and η1 characterize evolved nominal distributions at time T from any pair of initial conditions at some earlier time t =

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48. [48]

    If we use the stationary-path conditions to propagate values of θ and η through time, and define V (τ ) to be the area inside the image of the rectangle in Eq. ( 129) along these stationary trajectories, time-invariance of t he inner product, and the Liouville conservation of volume elements in dual coordinates, implies that d dτ ∫ V dθn dη = 0 (131) Note ...

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49. [49]

    ( 122) as d dτ ∫ V dv du ˆΦ2 0 = 0 (132) which is the conserved integral graphed in Fig 1

    could be recast using Eq. ( 122) as d dτ ∫ V dv du ˆΦ2 0 = 0 (132) which is the conserved integral graphed in Fig 1. In Eq. ( 132) ˆΦ2 0, the 2-dimensional differential of the scaled CGF ψ/N =− log ˆΦ0, appears explicitly as the density of overlap of dv with du that, like ψ itself, is con- stant along stationary paths. ˆΦ2 0 is not independent of the posit...

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50. [50]

    ( 129) are not, so formally the range of the sample estimator ( 130) remains unbounded over any duration T

    are bounded, the limits on (η2− η1) in Eq. ( 129) are not, so formally the range of the sample estimator ( 130) remains unbounded over any duration T . However, for any fixed values of (nB− nA)t=T and starting uncertainty (η2− η1)t=0, the total information obtainable from large-deviations sampling about differ- ences in the initial conditions is finite and d...

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51. [51]

    The embedding for general distributions on finite state spaces Eq. (

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52. [52]

    Suppose |{n}|is finite in order illustrate the Fisher embedding geometry for distri- butions over finite state spaces

    in the text can be written in the form δs2 = 4δθiδθj ∑ n ∂ √ ˜ρ(θ) n ∂θ i ∂ √ ˜ρ(θ) n ∂θ j (A1) Let|{n}|be the cardinality of the set of states on which ρn is defined (for example, in chemistry, only a sub-lattice of all integer-valued vectors in the positive orthant may ever be accessible as counts, given a system’s stoichiom- etry and conserved quantitie...

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53. [53]

    Embeddings in reduced dimension for exponential families on the multinomial The Poisson (

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54. [54]

    All fac- torial moments are powers their first moments, causing the CGF for many particles to scale as a multiple of a single-particle CGF

    and multinomial ( 14) distributions are both in a class recognized by Anderson, Craciun, and Kurtz (ACK) [26] in connection with uniqueness of sta- tionary solutions for chemical reaction networks. All fac- torial moments are powers their first moments, causing the CGF for many particles to scale as a multiple of a single-particle CGF. It is not then surpr...

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55. [55]

    for the Fisher metric in terms of a distri- bution ˜ρn with possibly indefinitely many independent terms, for the ACK distributions projects to a function of the same form in terms of expected numbers ni over the D independent species. To see how this works for a distribution ρ(n0) n with multinomial form ( 14), express the expected number frac- tions as n...

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56. [56]

    becomes δs2 = D∑ j=1 δn2 j nj (A10) which is the same function of n as the function of ˜ρ in the third line of Eq. ( 6). Appendix B: Sample means and variances in the large-deviation approximation to threshold indicator expectations The expectation of the tilted indicator function from Eq. ( 27) may be written in a series of inequalities culmi- nating in ...

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57. [57]

    in the text. 25 To estimate the tightness of the bounds, begin by ob- serving that in the large-deviation scaling regime ( 31), with all cumulants generated as derivatives of ψ∼ N , the expansion of central moments in terms of cumulants bounds the scaling of the kth central moment as ⣨ (n−⟨ n⟩)k ⟩ /⟨n⟩k≤O ( N −⌈ k/2⌉ ) (B3) The log ratio we wish to bound ...

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58. [58]

    Doi operator algebra and inner product The main constructs in the Doi operator formula- tion [2, 3] of moment-generating functions as formal power series are as follows: The identification (

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59. [59]

    (C1) to stand for the commutator algebra between components of ∂/∂z and factors of z, applied by function composition acting to the right on MGFs

    of z and ∂/∂z with raising and lowering operators a† and a allows the commutator alge- bra [ ai, a† j ] = δij. (C1) to stand for the commutator algebra between components of ∂/∂z and factors of z, applied by function composition acting to the right on MGFs. Monomials zn from Eq. (

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60. [60]

    are basis elements in a lin- ear space of MGFs, built up by multiplication on the number 1. A bracket notation for states and an inner product are introduced by the pair of denotations 1→| 0) ∫ dDz δD(z)→ (0| (C2) Each monomial zn is denoted as a number state D∏ i=1 zni i × 1→ D∏ i=1 a† i ni |0)≡| n) . (C3) The number states are eigenstates of the set of ...

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61. [61]

    Coherent states and Peliti functional integral The uniform measure in the 2D-dimensional integral for the representation of unity (

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62. [62]

    in the main text is known as the Haar measure. Using the definition ( 40) for coherent states and ( 41) for their dual projectors, and expanding the exponential functions as sums, ∫ dDφ†dDφ πD |φ) (φ| = ∫ dDφ†dDφ πD D∏ i=1 e− φ† i φi ∑ ni ∑ mi φni i φ† i mi ni! |n) (m| = ∑ n |n) (n|= I (C8) The phase component in each integral dφ† i dφi vanishes unless ni ...

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63. [63]

    and elsewhere. Appendix D: Stationary-point approximations to the Wigner function The functional integral provides the most direct route to the current conservation law ( 60) for the Wigner func- tion. It is possible, with somewhat more work, to derive 13 There is a notational subtlety in writing the complex area integral ∫ dφ† dφ ≡ ∫ ∞ 0 d |φ| ∫ 2π 0 |φ|...

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64. [64]

    returns the analytic representation of the MFG: Ψ(z) = (0|eza|Ψ) = ∫ dDφ (0|eza|φ) ρ(φ) = ∫ dDφ e(z− 1)φρ(φ) (D2) Now evaluate the integral in Eq. ( 57) at time t→ T , where the stationary value of the field φ† will coincide with the imposed argument z, wT ( φ† ‡, φ‡ ) = 1 πD ∫ dDφ e(φ† ‡ − z)(φ− φ‡ )e(z− 1)φρ(φ) (D3) It follows then that ∫ dDφ† ‡ wT ( φ† ...

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65. [65]

    and ( D2). Stationary-point approximations The stationary value ¯φ(z) of the tilted density e(z− 1)φρ(φ) is given by ∂ log ρ ∂φ ⏐ ⏐ ⏐ ⏐ ¯φ(z) = 1− z (D5) The two stationarity conditions on the arguments of wT follow from Eq. ( D3) as ∂ ∂φ‡ wT ( φ† ‡, φ‡ )⏐ ⏐ ⏐ ⏐ φ† ‡ =z = ( z− φ† ‡ ) wT (z, φ‡) ∂ ∂φ† ‡ wT ( φ† ‡, φ‡ )⏐ ⏐ ⏐ ⏐ ⏐ φ† ‡ =z = ∫ dDφ (φ− φ‡) e(z−...

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66. [66]

    Stationary-path equations and final-time conditions for response fields The stationary-path equations of motion for the com- ponents of the field φ† from Eq

    Coherent-state and number-potential solutions a. Stationary-path equations and final-time conditions for response fields The stationary-path equations of motion for the com- ponents of the field φ† from Eq. ( 47), in the rotated ba- sis ( 98), evaluate to ∂τ Φ† = ∂ ˆL ∂ ˆΦ =− νφ† ∂τ φ† = ∂ ˆL ∂ ˆφ = φ† (E1) The final-time values φ† T are given by variation of...

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67. [67]

    ( 111) in the main text

    requires varying z along the contour za≡ νT a νa zb≡ ν T b νb (E3) giving Eq. ( 111) in the main text. The remaining time- dependent solutions, with time argument denoted explic- itly here by subscript τ , are given by φ† τ = (ν T− ν)(1 4− ν 2)eτ − T Φ† τ = 1− νφ† τ (E4) Initial data are specified in the generating function ψ0, which when evaluated at the ...

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68. [68]

    Variation of this term against ψ0 with respect to φ† 0 gives the initial-value conditions for the components of ˆφ0, as ˆφ0a = ∂ψ0 ∂φ† 0a ˆφ0b = ∂ψ0 ∂φ† 0b (E7) Solutions to the equations of motion ( E6) from these ini- tial conditions are then ˆΦt = ˆΦ0 = 1 1 + Λe− T ˆφt = ˆΦ0 [ ν + (¯ν0− ν) e− τ ] (E8) The two displacements defined in equations ( 102) an...

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69. [69]

    It follows from Eq

    in the text. It follows from Eq. ( E9) and Eq. ( E10) that the combination (ντ− ν) (¯ντ− ν)(1 4− ν 2) = Λe− T (E11) is invariant at its initial value. The CGF for a binomial distribution at any time re- tains the form ( 90), with ν a/νa and νb/νb replacing za and zb, and ¯νa and ¯νb replacing νa, and νb. From Eq. ( E5) and the invariant form ( E11), it fo...

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70. [70]

    Fisher spherical embedding In one dimension, the Pythagorean theorem (

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71. [71]

    The mean-value Fisher-sphere construction of Sec

    for K-L divergences loses the interpretation of a direction co- sine between vector fields, but still reflects scale changes between coherent-state or exponential families and the geometric coordinate. The mean-value Fisher-sphere construction of Sec. A 2, for one variable, is the embedding on a circle: cos2 α≡ 1 2 + ν sin2 α≡ 1 2− ν (E14) The coordinate di...

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72. [72]

    becomes ˆΦ2 0 = 1 [1 + uv]2 = (1 4− ν2) (1 4− ν2) (1 4− ¯ν2) = 4 (∂α/∂θ ) (∂α/∂η ) (∂v/∂θ ) (∂u/∂η ) = 4 ∂α ∂v ∂α ∂u (E17) the invariant Fisher information in coherent-state coor- dinates. The equivalence between the two forms ( 67) and ( 69) for the Fisher metric is again recovered as ∂2 ∂θ 2 ( ψ N ) = 4 ∂α ∂θ ∂α ∂η (E18) showing the variation of the emb...

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73. [73]

    The measure changes (dv/dθ) and (du/dη) are the same as those in the Fisher metric, but in addition to these the term u + v is not invariant

    Evaluation of the Amari-Chentsov tensor From the form (120) of the Fisher metric in coordinates (v, u), the Amari-Chentsov tensor on all contravariant indices can be computed: T N ≡ ∂3 ∂θ 3 ( ψ N ) =− 2 (dv/dθ) (du/dη) [1 + uv]3 [ ν (1− uv) + √ 1 4− ν 2 (u + v) ] (E19) T is symmetric under u↔ v, but its magnitude is not conserved along the stationary-path...

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74. [74]

    ( 93) and η of Eq

    Connection coefficients in the coherent-state connection Among the nonzero connection coefficients for the dual coherent-state connections ( 88), the only independent components are for the rotated variables θ of Eq. ( 93) and η of Eq. ( 113). They are Γ(θ) θθ θ = ∂ ∂θ log (∂ν ∂θ ) Γ(η)∗ ηη η = ∂ ∂η log (∂ ¯ν ∂η ) (E20) The covariant components of the time de...

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75. [75]

    There- fore the second derivative ∂2ψ∗ ∂n ∂η =− ∂θ ∂n ⏐ ⏐ ⏐ ⏐ η ∂2ψ ∂θ ∂η =− ∂2ψ∗ ∂n2 ∂2ψ ∂θ ∂η =− (∂2ψ ∂θ 2 )− 1 ∂2ψ ∂θ ∂η =− 1 (E26) The third line of Eq

    T wo-dimensional divergences of the large-deviation function and their integrals In the exponential family of tilted distributions with tilting parameter θ, over a base distribution with expo- nential parameter η, the large-deviation function of two arguments is constructed as ψ∗ (n; η) = θ(n; η) n− ψ(θ(n; η) ; η) (E24) Its variation with η at fixed n is g...

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