REVIEW 3 major objections 3 minor 27 references
Derivation of Fokker-Planck equation from Schrodinger dynamics
T0 review · 3 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash
Pith's one-line read This paper claims that the Fokker–Planck equation can be derived from the reversible Schrödinger equation by neglecting the coherence terms in the occupation-probability decomposition, showing unitary quantum evolution as the source of…
desk verdict The paper restates the textbook Pauli-master-equation route and then hand-waves away quantum coherences; the central step is unsupported, and a basic time-index error in Eq. (9) breaks the derivation from the start. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The central object is the decomposition $P_k(t)=\sum_l T_{kl}P_l(t)+\sigma_k(t)$ of the occupation probability in the $H_0$ eigenbasis, together with the transition rate $W_{kl}=(T_{kl}-\delta_{kl})/\Delta t$. The coherence terms $\sigma_k(t)$ are the off-diagonal interferences that preserve unitarity; neglecting them converts the reversible Schrödinger evolution into a Markovian balance equation, and discrete derivatives in the state index carry that equation to the diffusion and Fokker–Planck limits. In the path-integral section, the machinery is the quantum propagator $\langle k|\exp(-iHt/\hbar)|k'\rangle$, split into short-time kinetic and potential factors whose iteration yields a path integral in which the Fokker–Planck action involves response variables $\tilde{k}$ conjugate to $k$.
What would settle it
For a finite quantum system such as a two-level or two-site model, compute the exact time evolution of $P_k(t)$ from the Schrödinger equation and compare it with the solution of the master equation (18); if the coherence term $\sigma_k(t)$ remains comparable to the Markovian term during the times of interest, the derivation's central neglect fails and the predicted diffusion does not occur.
Extended reading notes
Core claim
The central claim is that a classical diffusive limit is already present inside the reversible Schrödinger equation. Expanding the state in the eigenbasis $|k\rangle$ of the time-independent part $H_0$ of the Hamiltonian gives $P_k(t)=\sum_l T_{kl}P_l(t)+\sigma_k(t)$, with $P_k=|a_k|^2$, transition probabilities $T_{kl}=|U_{kl}|^2$ from the unitary evolution operator, and coherence terms $\sigma_k(t)$ built from the off-diagonal amplitude products. Neglecting $\sigma_k$ closes the equations into the Pauli master equation $\partial_t P_k=\sum_{l\neq k}(W_{kl}P_l - W_{lk}P_k)$; under detailed balance and with discrete second derivatives this becomes the diffusion equation, and allowing a drift term and position-dependent diffusivity produces the generalized Fokker–Planck equation (25). The paper concludes that the path-integral representation of the Fokker–Planck equation emerges from the same propagator used in quantum mechanics, and that the thermodynamic entropy defined on the occupation probabilities corresponds to a coarse-graining of the constant von Neumann entropy.
Load-bearing premise
The load-bearing assumption, acknowledged in the concluding paragraph, is that the state index $k$ refers to eigenstates of $H_0$ rather than of the full $H(t)$, and that the coherence terms $\sigma_k(t)$ can be neglected; the paper gives no physical mechanism that would make those terms small.
Editorial extensions
If this is right
- If the derivation is correct, irreversible diffusion emerges from unitary Schrödinger dynamics without any additional statistical postulate: discarding the off-diagonal interference terms is enough to produce a Markovian rate equation.
- The transition rates $W_{kl}$ are not phenomenological inputs; they are fixed by the matrix elements of the unitary evolution operator in the $H_0$ eigenbasis.
- The Fokker–Planck equation inherits a quantum path-integral representation, so field-theoretic solution techniques carry over directly to stochastic processes of this type.
- The thermodynamic entropy $S=-\sum_k P_k\ln P_k$ is a coarse-grained version of the von Neumann entropy, which explains how entropy can increase even though the underlying unitary evolution preserves the full quantum entropy.
Reading between the lines
- A direct numerical test is to compute $\sigma_k(t)$ for a concrete finite Hamiltonian and compare its size with the Markovian term; for many closed systems the coherence terms oscillate rather than decay, so the domain of validity of the derivation is not automatically large.
- The paper leaves open the physical mechanism that would justify neglecting $\sigma_k$; identifying such a mechanism, such as decoherence, weak coupling, coarse-graining, or measurement, is the natural next step and would turn the conditional result into a predictive statement.
- The same algebra suggests that coupling the system to an environment rather than discarding coherences would produce an open-quantum-system master equation whose limit reproduces the Fokker–Planck equation, connecting the derivation to standard open-system theory.
- Because the result depends on expanding in the eigenbasis of $H_0$, the resulting Fokker–Planck description is basis-dependent; whether a more invariant formulation exists is untested.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript claims to derive the Fokker-Planck equation from the reversible Schrödinger equation by expanding the wavefunction in eigenstates of H0, separating diagonal occupation probabilities from coherence terms, and neglecting the coherence terms σ_k to obtain a Markovian master equation. From that master equation, via discrete derivative approximations, the author obtains the diffusion equation and then the generalized Fokker-Planck equation. The paper also includes a standard derivation of the path integral representation of the Fokker-Planck equation and concludes that the thermodynamic entropy is a coarse-grained quantum entropy in the H0 eigenbasis.
Significance. If the derivation were correct, it would address a longstanding question of how irreversible classical diffusion emerges from unitary quantum dynamics. The paper correctly identifies the coherence terms σ_k as the obstruction to classical behavior and honestly states the key assumption of using H0 eigenstates at the end. However, the central derivation is not supported by the mathematics: there is a concrete error in the time argument of the expansion coefficients, the neglect of σ_k is unjustified and effectively postulates classical Markovianity, and the transition rates as defined vanish in the small-time limit for unitary evolution. No machine-checked proofs, reproducible code, or independent numerical checks are provided. The significance of the claimed result is therefore not realized by the manuscript in its current form.
major comments (3)
- [§2, Eqs. (9)–(13)] Equation (9) is incorrect: from |ψ(t)⟩ = U|ψ(0)⟩ and the expansion in the H0 eigenbasis, one must obtain a_k(t) = Σ_l U_kl a_l(0), not Σ_l U_kl a_l(t) as written. This error propagates into Eq. (13), where the diagonal term becomes Σ_l T_kl P_l(t) instead of Σ_l T_kl P_l(0). Consequently Eq. (13) is not a valid identity for the exact Schrödinger dynamics, and the subsequent master equation (18) is built on this erroneous relation.
- [§2, Eqs. (13)–(18)] Even if Eq. (9) were corrected, the passage to the Markovian master equation is not justified. For a unitary U, T_kl − δ_kl = O(Δt²) for l ≠ k, so the rate W_kl defined in Eq. (17) vanishes as Δt → 0; no non-trivial continuous-time rates exist without an additional coarse-graining or weak-coupling/Fermi-golden-rule limit, and none is provided. A concrete counterexample is a degenerate two-level system with V = λ(|1⟩⟨2| + |2⟩⟨1|) prepared in |1⟩: here σ_k = 0 exactly, yet P_1(t) = cos²(λt/ℏ) oscillates, which cannot be reproduced by Eq. (18) with constant rates. Thus the neglect of σ_k and the Markovianization together assume the classical behavior the paper purports to derive.
- [§3, Eqs. (20)–(24)] The discrete derivative approximations are not valid as stated. Equation (23) uses [P_k(t) − P_k(0)]/Δt, which is a backward difference from the initial time, not the derivative at time t required for a differential equation. Equation (24) mixes P_{k±l}(t) with P_k(t) without justifying the continuum limit or the convergence of the infinite sum in Eq. (21). No derivation is given that Eq. (18) under detailed balance actually reduces to the diffusion equation (20). This unproven continuum limit is the bridge to the Fokker-Planck equation (25), so the central claim of the manuscript is not established.
minor comments (3)
- [§4, Eq. (40)] The prefactor in the time-sliced path integral is incorrect: the standard form has a factor (m/(2πiℏΔt))^{N/2} with N factors and N−1 integrals, whereas Eq. (40) has a single factor m/(2πiℏΔt) outside the integral. As written, the expression is dimensionally inconsistent and does not reduce to Eq. (38) in the free-particle limit.
- [§4, Eq. (44)] The contour orientation in Eq. (44) appears to be incorrect: the standard Fourier representation of the delta function with this integrand requires the contour from −i∞ to i∞; the orientation written here would introduce an overall sign, propagating into Eq. (45).
- [Conclusion, Eq. (47)] The statement that the entropy S defined in Eq. (47) 'must always increase' is not proven in the manuscript. The master equation (18) alone does not guarantee monotonic increase of the Shannon entropy without additional conditions on the transition rates and the initial distribution; this assertion goes beyond what the derivation establishes.
Circularity Check
The Markovian master equation is assumed by construction: Eq. (13) contains P_l(t) on the right-hand side, so dropping sigma_k and rewriting via Eq. (17) merely restates the Markovian balance as Eq. (18).
-
self definitional
[Eqs. (9), (13), (17), (18) and the paragraph following Eq. (13), pp. 1-2]
"a_k(t) = Σ_l U_{kl} a_l(t) (9) ... P_k(t) = Σ_l T_{kl} P_l(t) + σ_k(t) (13) ... Therefore, if the σ_k terms could be neglected in Eq. (13), the time evolution of the occupation probability would be described by a classical Markovian process ... ∂P_k/∂t = Σ_{l≠k} [W_{kl}P_l − W_{lk}P_k] (18)."
Since |ψ(t)⟩ = U|ψ(0)⟩, the coefficient in Eq. (9) must be a_l(0), not a_l(t). With the correct argument, the diagonal part of P_k(t) is Σ_l T_{kl}P_l(0), not Σ_l T_{kl}P_l(t). The paper's time-local form makes Eq. (13) with σ_k=0 already the one-step Markovian balance equation, and Eq. (18) is only its differential rewrite via Eqs. (17) and (23). No mechanism from unitary Schrödinger dynamics produces this Markovian semigroup structure; it is inserted by construction through the wrong time argument. Hence the derived FP equation is equivalent to the assumed Markovian balance, making the central claim circular.
full rationale
The derivation chain from Schrödinger dynamics to the Fokker-Planck equation is not self-contained: the central Markovian step is built into the equations rather than obtained from unitarity. In particular, Eq. (9) writes a_k(t)=Σ_l U_{kl}a_l(t) although U evolves the initial coefficients, so Eq. (13) acquires P_l(t) on the right-hand side; dropping σ_k then yields a classical Markovian balance by definition, and Eq. (18) is its rate form. The later diffusion and Fokker-Planck equations (20)-(25) are standard generalizations of this assumed master equation. There are no load-bearing self-citations: the references are standard textbooks and unrelated works, and no uniqueness theorem or prior result by the author is invoked to forbid alternatives. The paper openly labels the eigenbasis choice as the key assumption, but this is not the only hidden input; the Markovian property itself is the circular element. Because the central claim reduces by construction to the time-local Markovian assumption, the circularity score is high.
Assumptions & free parameters
assumptions (4)
- ad hoc to paper The coherence terms σ_k in Eq. (13) can be neglected.
- domain assumption The index k corresponds to eigenstates of the time-independent Hamiltonian H0, not the full Hamiltonian H(t).
- standard math The transition probabilities T_kl are differentiable in time, allowing the definition of transition rates W_kl.
- ad hoc to paper The discrete derivative approximations in Eqs. (23)-(24) are valid in the limit of interest.
Cite this review
Pith. "Pith review of Derivation of Fokker-Planck equation from Schrodinger dynamics." pith.science (2026). https://pith.science/paper/CLX73KHJ
@misc{pith2026250522693,
author = {Pith},
title = {Pith review of: Derivation of Fokker-Planck equation from Schrodinger dynamics},
year = {2026},
howpublished = {\url{https://pith.science/paper/CLX73KHJ}},
note = {Machine review of arXiv:2505.22693}
}
read the original abstract
The Fokker_Planck equation can be derived in a consistent manner through a microscopic approach based on a unified scheme of classical and quantum mechanics. Here we shall derive it through a purely quantum mechanical approach based on the reversible Schrodinger dynamics. We also give a brief discussion of the path integral representation of the Fokker_Planck equation in light of our derivation. We conclude that, because of the use of the representation of eigenstates of the time-independent Hamiltonian in our derivation, the thermodynamical entropy in this case must correspond to a coarse-graining of the quantum entropy.
Reference graph
Works this paper leans on
-
[1]
E. A. Calzetta and B-L. B. Hu, Nonequilibrium Quantum Field Theory (Cambridge University Press, 2008)
work page 2008
- [2]
-
[3]
(21) Thus the simplest form of the Fokker-Planck equation is the diffusion equation
may be written as, ∂Pk ∂t = D 2 ∂2Pk ∂k 2 , (20) which is the diffusion equation with the diffusion coeffi- cient defined as D = 2 ∞∑ l=1 Wk,k+ll2. (21) Thus the simplest form of the Fokker-Planck equation is the diffusion equation. If the initial condition is taken as Pk(0) = δ(k), where δ(k) is the Dirac delta function, the solution is given by Pk(t) = 1 √ 2π...
- [4]
-
[5]
to |ψ(t)⟩ = exp ( −i ℏHt ) |ψ(0)⟩. (6) Equation ( 6) is more concisely written as |ψ(t)⟩ =U |ψ(0)⟩ (7) whereU is the unitary time evolution operator satisfying UU † =I (8) and I is the identity matrix. From Eqs. ( 3) and ( 7) one can obtain ak(t) = ∑ l Uklal(t) (9) where Ukl ≡ ⟨k|U |l⟩, and where ∑ k ⟨k|k⟩ = 1 (10) has been used since the total probabilit...
-
[6]
(28) This propagator is the central object of Feynman’s for- mulation of quantum mechanics [17]
one can write ⟨k|ψ(t)⟩ = ∫ dk′⟨k| exp ( −i ℏHt ) |k′⟩⟨k′|ψ(0)⟩ (26) Or ψk(t) = ∫ dk′K(k,t,k ′, 0)ψk′ (0) (27) with the propagator K(k,t,k ′, 0) = ⟨k| exp ( −i ℏHt ) |k′⟩. (28) This propagator is the central object of Feynman’s for- mulation of quantum mechanics [17]. It contains the complete information about the eigenenergies En and the corresponding eig...
-
[7]
shall not hold. Since we have used the eigenstates of H0, and not of H(t), the thermodynamical entropy [19], which must always increase, S ≡ − ∑ k Pk lnPk (47) corresponds to a coarse-graining of the quantum entropy S ≡ − Tr (ρ lnρ). (48) which in this case must stay constant, where ρ denotes the density matrix of the system [20]. ∗ Correspondence. lone.i...
-
[8]
Paul and J
W. Paul and J. Baschnagel, Stochastic Processes: From Physics to Finance (Springer, 2013)
2013
Show all 27 references
-
[9]
T. D. Frank, Phys. Rev. E 71, 031106 (2005)
2005
-
[10]
D. J. Evans, S. J. Searles, and L. Rondoni, Phys. Rev. E 71 056120 (2005). 5
2005
-
[11]
F. M. Fern´ andez,Phys. Scr. 80, 065010 (2009)
2009
-
[12]
Colangeli and L
M. Colangeli and L. Rondonia, Physica D 241 681 (2012)
2012
-
[13]
N. N. Bogolyubov Jr. and D. P. Sankovich, Russian Math. Surveys 49 19 (1994)
1994
-
[14]
T. D. Frank, Nonlinear Fokker-Planck equations: funda- mentals and applications (Springer Science & Business Media, 2005)
2005
-
[15]
Rice, An Introduction to Quantum Optics (IOP Pub- lishing Ltd, 2020)
P. Rice, An Introduction to Quantum Optics (IOP Pub- lishing Ltd, 2020)
2020
-
[16]
Zinn-Justin, Quantum field theory and critical phe- nomena
J. Zinn-Justin, Quantum field theory and critical phe- nomena. (Clarendon Press Oxford, 1996)
1996
-
[17]
J. M. Heninger, D. Lippolis, and P. Cvitanovi´ c,Commun. Nonlinear Sci. Numer. Simulat. 55, 16 (2018)
2018
-
[18]
is sometimes also expressed as ∂Pk ∂t = ∑ k MklPl(t), (19) where the Mkl denote the Markov transition matrix el- ements describing the transition rates between sites k and l [15]. The above Markov process is said to satisfy detailed balance provided there exists a stationary d...
-
[19]
Zhou, M-G Li, and S
T. Zhou, M-G Li, and S. Yan, Phys. Rev. C 111, 044001 (2025)
2025
-
[20]
L. E. Ballentine, Quantum Mechanics: A Modern Devel- opment (World Scientific Publishing Company, 2014)
2014
-
[21]
H. P. Breuer, E. M. Laine, J. Piilo, and B. Vacchini, Rev. Mod. Phys. 88, 1 (2016)
2016
-
[24]
R. P. Feynman, Rev. Mod. Phys. 20, 367 (1948)
1948
-
[25]
(43) where the k-derivatives here act only on the delta func- tion and not on Pk(t)
yields ∂Pk′ (t) ∂t = − ∂ ∂k ′ [µk′ (t)Pk′ (t)] + ∂2 ∂k ′2 [Dk′ (t)Pk′ (t)] = ∫ ∞ −∞ dk [( µk(t) ∂ ∂k +Dk(t) ∂2 ∂k 2 ) δ(k′ −k) ] Pk(t). (43) where the k-derivatives here act only on the delta func- tion and not on Pk(t). Integrating over a time interval ǫ gives Pk′ (t +ǫ) = ∫ ...
-
[26]
Kleinert, Path Integrals in Quantum Mechanics, Statistics and Polymer Physics (World Scientific 1995)
H. Kleinert, Path Integrals in Quantum Mechanics, Statistics and Polymer Physics (World Scientific 1995)
1995
-
[27]
Alicki and M
R. Alicki and M. Fannes, Lett. Math. Phys. 32 75 (1994)
1994
-
[28]
M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, 2010)
2010
-
[34]
into a part dependent on the kinetic energy and another part on the potential energy. From an expansion of the Baker- Hausdorff formula we find exp ( −i ℏ (T +V )∆ t ) ≈ exp ( −i ℏT ∆ t ) exp ( −i ℏV ∆ t ) + 1 ℏ2 [T,V ](∆ t)2 where we have neglected terms of order (∆ t)3 and hig...
Reviewed August 7, 2026 · model on record in the stance chip above.
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