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Non-Markovian Mesoscopic Dynamics within the Energy Distribution Framework: A Generative Master Equation and the Unification of Transport Theories

Ende Guo

Published Mar 11, 2026• 0 citations
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Abstract

This paper achieves a fundamental reconstruction of the quantum theory paradigm based on the energy distribution framework. This framework radically steps outside the traditional description based on probability amplitude evolution, reinterpreting physical processes as deterministic energy flow among quantum units ($Q$), an interfacial defect network ($D$), and bulk dissipation channels ($B$). A series of previous works have laid a solid foundation for this new paradigm: the first paper rigorously proved the exact relation between internal dissipation power and energy relaxation rate, providing a mathematical link for the energy perspective; the second paper established a single-body energy distribution theory, defined the core order parameter—the energy distribution ratio $f$, and proved that the relaxation rate is entirely determined by the microstructure of $D$; the third paper extended the theory to many-body systems and, under the Markovian approximation, derived a mesoscopic diffusion equation describing energy transport within the $D$ network. As the fourth paper in this series, this work completely abandons the Markovian approximation and constructs a universal theory of non-Markovian mesoscopic dynamics from first principles. Taking the energy density $u(\mathbf{r},t)$ as the fundamental field variable, this paper derives an integro-differential equation containing memory kernels $\mathcal{K}_0$ and $\mathcal{K}_2$ by tracing over the degrees of freedom describing the defect network $D$ and implementing spatial coarse-graining—termed herein the \textbf{generative master equation}. The essence of this equation lies in the fact that the memory kernels are completely determined by the spatial correlation function $\mathcal{C}_D^{(+)}(s,t)$ of the defect network $D$, which, as the system's "microscopic fingerprint," uniquely encodes all dynamical information of $D$. By selecting different forms of the correlation function, the master equation can naturally reduce to or transform into almost all known macroscopic transport equations in physics: in the Markovian limit, it returns to the diffusion equation; exponential correlation yields the telegraph equation and a finite propagation speed $v=\sqrt{D_0/\tau_c}$; power-law correlation gives rise to fractional diffusion equations; nonlinear correlation leads to nonlinear diffusion equations. This unified mapping reveals the deep-seated role of the theory presented herein: it does not seek to replace existing theories but rather to provide them with a common microscopic origin. This paper also establishes a quantitative criterion for energy backflow and directly links it to the energy distribution ratio $f$, opening new avenues for the experimental inversion of memory kernels. This work marks the evolution of the energy distribution framework from a specific model into a generative mother theory capable of uniformly describing various transport behaviors. \textbf{Keywords}: Generative master equation; Energy distribution framework; Non-Markovian dynamics; Mesoscopic dynamics; Memory kernel; Green's function; Telegraph equation; Finite propagation speed

DOI
10.5281/zenodo.18945819

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quant-pharXiv:2605.29508•May 29, 2026•0 citations
This paper achieves a fundamental reconstruction of the quantum theory paradigm based on the energy distribution framework. This framework radically steps outside the traditional description based on probability amplitude evolution, reinterpreting physical processes as deterministic energy flow among quantum units (QQQ), an interfacial defect network (DDD), and bulk dissipation channels (BBB). A series of previous works have laid a solid foundation for this new paradigm: the first paper rigorously proved the exact relation between internal dissipation power and energy relaxation rate, providing a mathematical link for the energy perspective; the second paper established a single-body energy distribution theory, defined the core order parameter—the energy distribution ratio fff, and proved that the relaxation rate is entirely determined by the microstructure of DDD; the third paper extended the theory to many-body systems and, under the Markovian approximation, derived a mesoscopic diffusion equation describing energy transport within the DDD network. As the fourth paper in this series, this work completely abandons the Markovian approximation and constructs a universal theory of non-Markovian mesoscopic dynamics from first principles. Taking the energy density u(r,t)u(\mathbf{r},t)u(r,t) as the fundamental field variable, this paper derives an integro-differential equation containing memory kernels K0\mathcal{K}_0K0​ and K2\mathcal{K}_2K2​ by tracing over the degrees of freedom describing the defect network DDD and implementing spatial coarse-graining—termed herein the \textbf{generative master equation}. The essence of this equation lies in the fact that the memory kernels are completely determined by the spatial correlation function CD(+)(s,t)\mathcal{C}_D^{(+)}(s,t)CD(+)​(s,t) of the defect network DDD, which, as the system's "microscopic fingerprint," uniquely encodes all dynamical information of DDD. By selecting different forms of the correlation function, the master equation can naturally reduce to or transform into almost all known macroscopic transport equations in physics: in the Markovian limit, it returns to the diffusion equation; exponential correlation yields the telegraph equation and a finite propagation speed v=D0/τcv=\sqrt{D_0/\tau_c}v=D0​/τc​​; power-law correlation gives rise to fractional diffusion equations; nonlinear correlation leads to nonlinear diffusion equations. This unified mapping reveals the deep-seated role of the theory presented herein: it does not seek to replace existing theories but rather to provide them with a common microscopic origin. This paper also establishes a quantitative criterion for energy backflow and directly links it to the energy distribution ratio fff, opening new avenues for the experimental inversion of memory kernels. This work marks the evolution of the energy distribution framework from a specific model into a generative mother theory capable of uniformly describing various transport behaviors. \textbf{Keywords}: Generative master equation; Energy distribution framework; Non-Markovian dynamics; Mesoscopic dynamics; Memory kernel; Green's function; Telegraph equation; Finite propagation speed
Important Update on March 9, 2026: In this version, the description of environmental effects has been corrected: they are now properly treated as encapsulated within the effective spectral density of the interfacial defect network D, rather than as direct driving terms. All core results and conclusions remain unchanged, but the physical interpretation is refined to align with the energy distribution framework introduced in Part II of this series. Version 2 This paper is the third in the "Interface-Projected Realism" series. This series of work aims to establish a new theoretical paradigm: Ontologically, replace probabilistic wave function descriptions with deterministic energy flow; Methodologically, redefine physical reality as an interactive whole composed of quantum units (QQQ), an interfacial defect network (DDD), and bulk channels (BBB). In the first work, an exact relationship between energy flow and relaxation rate was established, providing a formal bridge for the paradigm shift. In the second work, the "Energy Distribution Framework" was proposed, demonstrating that dissipation is the deterministic allocation of energy between QQQ and DDD, and deriving the universal formula for internal dissipation power Rint=ℏωq[Γ↓⟨n⟩−Γ↑(1−⟨n⟩)]R_{\mathrm{int}} = \hbar\omega_q[\Gamma_\downarrow\langle n\rangle - \Gamma_\uparrow(1-\langle n\rangle)]Rint​=ℏωq​[Γ↓​⟨n⟩−Γ↑​(1−⟨n⟩)], where the relaxation rates are completely determined by the internal structure of DDD, revealing the core role of the energy distribution ratio fff. This paper generalizes this new paradigm based on energy distribution from single-body to many-body systems. The core step is to define multiple QQQ contacts and their shared DDD and BBB together as a new, unified composite system. Within this framework, the traditionally complex "many-body correlated dissipation" is clearly interpreted as a macroscopic result of the deterministic flow of energy within this composite system, governed by allocation and transport rules determined by the structure of the DDD network. Through rigorous coarse-graining of this overall system, this paper derives the governing equation for the evolution of its internal energy density field u(r,t)u(\mathbf{r},t)u(r,t)—a classical reaction-diffusion equation:∂u∂t=Rtotal−γ0(r)u+∇⋅(γ2∇u).\frac{\partial u}{\partial t} = R_{\mathrm{total}} - \gamma_0(\mathbf{r}) u + \nabla \cdot (\gamma_2 \nabla u).∂t∂u​=Rtotal​−γ0​(r)u+∇⋅(γ2​∇u).Here, the local dissipation rate γ0(r)\gamma_0(\mathbf{r})γ0​(r) and the diffusion coefficient γ2\gamma_2γ2​ are completely determined by the properties of the DDD network, corresponding respectively to the spatial distribution of the energy distribution ratio fff from the second paper and the redistribution effect induced by its correlations. This theory is not only logically self-consistent but also provides a unified explanation for multiple cutting-edge experiments. This paper marks that a physical framework based on deterministic energy distribution and transport can now systematically describe many-body correlation phenomena, providing crucial support for the completeness of the new paradigm. \textbf{Keywords}: Interface-Projected Realism; Energy Distribution; Many-Body Systems; Correlated Dissipation; Reaction-Diffusion Equation Version 1 Traditional quantum theory typically attributes energy dissipation to random perturbations from the external environment. Based on the "Interfacial Projected Realism" framework, this work re-examines the nature of energy dissipation from a novel perspective, treating the qubit, the interfacial defect network, and the bulk dissipation channel as a self-contained total system. We first rigorously derive the universal formula for internal dissipation power and the energy distribution ratio fff in single-body quantum systems, and then extend these concepts to many-body systems. Starting from a microscopic Hamiltonian coupling multiple qubits to a common quantized field and employing the Nakajima-Zwanzig projection operator technique, we obtain a universal expression for the energy dissipated from a many-body system to the environment. Applying this result to the specific model of multiple two-level systems coupled to a common bosonic field yields a collective master equation where the dissipation coefficient matrix Γij\Gamma_{ij}Γij​ encodes all spatial and coupling information. Through systematic coarse-graining, gradient expansion, and semiclassical approximation, we derive for the first time from an energy-distribution perspective a closed set of macroscopic equations for the local system energy density u(r,t)u(\mathbf{r},t)u(r,t) and the environmental field energy density U(r,t)U(\mathbf{r},t)U(r,t): ∂u∂t=Rtotal(r,t)−γ0(U)u+∇⋅(γ2(U)∇u)\frac{\partial u}{\partial t} = R_{\mathrm{total}}(\mathbf{r},t) - \gamma_0(U) u + \nabla \cdot \bigl( \gamma_2(U) \nabla u \bigr)∂t∂u​=Rtotal​(r,t)−γ0​(U)u+∇⋅(γ2​(U)∇u)∂U∂t=D∇2U−ΓBU+ρ(r)γ0(U)u\frac{\partial U}{\partial t} = D \nabla^2 U - \Gamma_B U + \rho(\mathbf{r}) \gamma_0(U) u∂t∂U​=D∇2U−ΓB​U+ρ(r)γ0​(U)u Within this framework, the energy distribution ratio f(r,t)=γ0u/Rtotalf(\mathbf{r},t) = \gamma_0 u / R_{\mathrm{total}}f(r,t)=γ0​u/Rtotal​ emerges naturally as a macroscopic quantity, its physical essence being jointly determined by the system’s microscopic coupling structure (embodied by Γij\Gamma_{ij}Γij​) and quantum statistical state. The theory quantitatively explains several long-unexplained anomalous phenomena in cutting-edge experiments, demonstrates strong testability, and lays a solid foundation for exploring deeper connections between energy distribution and macroscopic physics. For the exact mathematical formulation and detailed derivations, please refer to the PDF file.
This paper is the fifth in the series of studies on the Energy Distribution Framework, serving as a pivotal nexus within the theoretical system. The Energy Distribution Framework reconceptualizes physical processes as deterministic energy flows among quantum units (QQQ), the interface defect network (DDD), and the internal environment (BBB). Previous works in the series have respectively established corresponding dynamical equations at the single-body, many-body, Markovian, and non-Markovian mesoscopic scales. Starting from the Hamiltonian of the total system and based on the QDB three-layer structure, this paper rigorously derives the \textbf{Generalized Microscopic Quantum Master Equation (GMQME)} (Eq. \eqref{eq:GME}) under non-Markovian conditions. The memory kernel of this equation is directly determined by the correlation function CD(+)(t)\mathcal{C}_D^{(+)}(t)CD(+)​(t) of the defect network DDD, providing a clear microscopic origin and a first-principles calculable path for the memory kernel. In the dissipationless limit (K=0\mathcal{K}=0K=0), the GMQME naturally reduces to the von Neumann equation, thereby establishing its status as a fundamental master equation in quantum theory. The GMQME exhibits powerful generative capacity: projecting onto the intrinsic state space yields the non-Markovian Liouville equation; Heisenberg decomposition leads to the generalized Langevin equation (revealing the microscopic origin of random forces); the Markovian approximation reduces it to the Lindblad equation; extending to two-body systems yields the dynamical equation for the correlation operator, revealing the off-diagonal term ΓAB\Gamma_{AB}ΓAB​ as the core driving force for entanglement generation, and reconstructs Bell's inequality; taking energy expectation values, applying the closure approximation, and coarse-graining projects onto the mesoscopic energy density master equation of the fourth paper, completing the theoretical loop from quantum microphysics to mesoscopic transport. This work signifies that the Energy Distribution Framework has formed a logically self-consistent and hierarchically distinct complete theoretical system, spanning from quantum foundations to mesoscopic transport. Crucially, the entire deductive chain of this paper proceeds from the total Hamiltonian to the Generalized Microscopic Quantum Master Equation (GMQME), and then to the non-Markovian Liouville equation, the generalized Langevin equation, the Lindblad equation, two-body correlation dynamics, and the mesoscopic energy density master equation—all based entirely on the intrinsic structure of the theory, without introducing any external artificial concepts (such as the discrete phase space Wigner function). This marks the achievement of genuine logical self-consistency and conceptual purity for the Energy Distribution Framework. Keywords:Paradigm shift; Energy Distribution Framework; Generalized Microscopic Quantum Master Equation (GMQME); Generalized Langevin equation; non-Markovian Liouville equation; Lindblad equation; non-Markovian dynamics; quantum entanglement; Interface Projection Realism; theoretical closure
Based on the "Interface Projection Realism" framework, this paper rigorously derives the universal formula for internal dissipation power and the microscopic expression for the energy distribution ratio f from first principles. We treat the qubit (Q), the interfacial defect network (D), and the bulk dissipation channel (B) as a self-contained total system, no longer presupposing any part of it as an external environment. By applying the Born-Markov approximation to the degrees of freedom of D and B, we obtain the reduced master equation for Q, and subsequently calculate the net energy flow rate from Q to the incoherent degrees of freedom—the internal dissipation power R_internal. The result is: R_internal = ħω_q [Γ↓ ⟨n⟩ - Γ↑ (1-⟨n⟩)], where Γ↓ and Γ↑ are transition rates determined by the Q-D coupling and the internal structure of D, and ⟨n⟩ is the excited state population of Q. This formula does not rely on any thermal equilibrium assumptions and is applicable to arbitrary Markovian non-equilibrium steady states. Building on this, we define the total effective power absorbed by the system as R_total = ⟨∂H_drive/∂t⟩ and introduce the system's stored energy E_stored = ⟨H_Q⟩. Under periodic averaging, the energy conservation equation reads dE_stored/dt = R_total - R_internal, where the overline denotes time average. The energy distribution ratio f = R_internal / R_total satisfies f = 1 in the periodic steady state where dE_stored/dt = 0; during transients, f can deviate from 1, its specific expression depending on microscopic parameters, reflecting the degree of deviation from the steady state. We further demonstrate that when D and B are in thermal equilibrium, the above formula reduces to the traditional fluctuation-dissipation theorem. This paper proves that the universal formula for internal dissipation power naturally contains the fluctuation-dissipation theorem under thermal equilibrium conditions, while its validity in non-equilibrium states constitutes a non-equilibrium generalization of the traditional theorem, providing a unified theoretical foundation for understanding the energetic origin of quantum decoherence.
The thesis comprises two major themes of quantum statistical dynamics. One is the development of quantum dissipation theory (QDT). It covers the establishment of some basic relations of quantum statistical dynamics, the construction of several nonequivalent complete second-order formulations, and the development of exact QDT. Another is related to the applications of quantum statistical dynamics to a variety of research fields. In particular, unconventional but novel theories of the electron transfer in Debye solvents, quantum transport, and quantum measurement are developed on the basis of QDT formulations. The thesis is organized as follows. In Chapter 1, we present some background knowledge in relation to the afore-mentioned two themes of this thesis. The key quantity in QDT is the reduced density operator ρ(t) ≡ trBρT(t); i.e., the partial trace of the total system and bath composite ρT(t) over the bath degrees of freedom. QDT governs the evolu-tion of reduced density operator, where the effects of bath are treated in a quantum statistical manner. In principle, the reduced density operator contains all dynam-ics information of interest. However, the conventional quantum transport theory is formulated in terms of nonequilibrium Green's function. The newly emerging field of quantum measurement in relation to quantum information and quantum computing does exploit a sort of QDT formalism. Besides the background of the relevant theoretical development, some representative experiments on molecular nanojunctions are also briefly discussed. In Chapter 2, we outline some basic (including new) relations that highlight several important issues on QDT. The content includes the background of nonequi-librium quantum statistical mechanics, the general description of the total compos-ite Hamiltonian with stochastic system-bath interaction, a novel parameterization scheme for bath correlation functions, a newly developed exact theory of driven Brownian oscillator (DBO) systems, and its closely related solvation mode trans-formation of system-bath coupling Hamiltonian in general. The exact QDT of DBO systems is also used to clarify the validity of conventional QDT formulations that involve Markovian approximation. In Chapter 3, we develop three nonequivalent but all complete second-order QDT (CS-QDT) formulations. Two of them are of the conventional prescriptions in terms of time-local dissipation and memory kernel, respectively. The third one is called the correlated driving-dissipation equations of motion (CODDE). This novel CS-QDT combines the merits of the former two for its advantages in both the application and numerical implementation aspects. Also highlighted is the importance of correlated driving-dissipation effects on the dynamics of the reduced system. In Chapter 4, we construct an exact QDT formalism via the calculus on path integrals. The new theory aims at the efficient evaluation of non-Markovian dis-sipation beyond the weak system-bath interaction regime in the presence of time-dependent external field. By adopting exponential-like expansions for bath corre-lation function, hierarchical equations of motion formalism and continued fraction Liouville-space Green's function formalism are established. The latter will soon be used together with the Dyson equation technique for an efficient evaluation of non-perturbative reduced density matrix dynamics. The interplay between system-bath interaction strength, non-Markovian property, and the required level of hierarchy is also studied with the aid of simple spin-boson systems, together with the three proposed schemes to truncate the infinite hierarchy. In Chapter 5, we develop a nonperturbative theory of electron transfer (ET) in Debye solvents. The resulting exact and analytical rate expression is constructed on the basis of the aforementioned continued fraction Liouville-space Green's func-tion formalism, together with the Dyson equation technique. Not only does it re-cover the celebrated Marcus' inversion and Kramers' turnover behaviors, the new theory also shows some distinct quantum solvation effects that can alter the ET mechanism. Moreover, the present theory predicts further for the ET reaction ther-modynamics, such as equilibrium Gibbs free-energy and entropy, some interesting solvent-dependent features that are calling for experimental verification. In Chapter 6, we discuss the constructed QDTs, in terms of their unified math-ematical structure that supports a linear dynamics space, and thus facilitates their applications to various physical problems. The involving details are exemplified with the CODDE form of QDT. As the linear space is concerned, we identify the Schrödinger versus Heisenberg picture and the forward versus backward propaga-tion of the reduced, dissipative Liouville dynamics. For applications we discuss the reduced linear response theory and the optimal control problems, in which the correlated effects of non-Markovian dissipation and field driving are shown to be important. In Chapter 7, we turn to quantum transport, i.e. electric current through molecular or mesoscopic systems under finite applied voltage. By viewing the nonequilibrium transport setup as a quantum open system, we develop a reduced-density-matrix approach to quantum transport. The resulting current is explicitly expressed in terms of the molecular reduced density matrix by tracing out the degrees of freedom of the electrodes at finite bias and temperature. We propose a conditional quantum master equation theory, which is an extension of the conven-tional (or unconditional) QDT by tracing out the well-defined bath subsets indi-vidually, instead of the entire bath degrees of freedom. Both the current and the noise spectrum can be conveniently analyzed in terms of the conditional reduced density matrix dynamics. By far, the QDT (including the conditional one) has only been exploited in a second-order form. A self-consistent Born approximation for the system-electrode coupling is further proposed to recover all existing nonlinear current-voltage behaviors including the nonequilibrium Kondo effect. Transport theory based on the exact QDT formalism will be developed in future. In Chapter 8, we study the quantum measurement of a qubit with a quantum-point-contact detector. On the basis of a unified quantum master equation (a form of QDT), we study the measurement-induced relaxation and dephasing of the qubit . Our treatment pays particular attention on the detailed-balance relation, which is a consequence of properly accounting for the energy exchange between the qubit and detector during the measurement process. We also derive a conditional quantum master equation for quantum measurement in general, and study the readout characteristics of the qubit measurement. Our theory is applicable to the quantum measurement at arbitrary voltage and temperature. A number of remarkable new features are found and highlighted in concern with their possible relevance to future experiments. In Chapter 9, we discuss the further development of QDT, aiming at an effi-cient evaluation of many-electron systems. This will be carried out by reducing the many-particle (Fermion or Boson) QDT to a single-particle one by exploring, e.g. the Wick's contraction theorem. It also results in a time-dependent density func-tional theory (TDDFT) for transport through complex large-scale (e.g. molecules) systems. Primary results of the TDDFT-QDT are reported. In Chapter 10, we summary the thesis, and comment and remark on the future work on both the theoretical and application aspects of QDT.
Traditional paradigms of energy transport rely fundamentally on the movement of material particles, the propagation of classical wave packets, or the excitation of localized quasi-particles through a medium. In this paper, we propose and rigorously formalize the theoretical framework of Particle-Free Energy Transport (PFET). By extending the microscopic definitions of energy flux operators in open quantum many-body systems, we decompose the total energy current into three distinct components: a particle flux contribution, a field propagation contribution, and a purely quantum correlation flux contribution. We demonstrate mathematically that under specific non-equilibrium quantum conditions, energy can be effectively transferred between spatially separated quantum systems even when both the particle flux and the classical field propagation vanish identically. This phenomenon relies entirely on the non-local dynamics of quantum correlation networks. We establish the governing master equations, derive the conservation laws associated with correlation-driven energy transfer, and discuss the thermodynamic consistency and fundamental implications of this paradigm for modern quantum physics.
We develop an interacting extension of the Double Covariance Model (DCM), a stochastic subquantum framework in which macroscopic quantum dynamics emerge through coarse-graining of correlated microscopic fluctuations. Starting from local stochastic differential equations on subsystem Hilbert spaces, we derive a closed evolution equation for a coarse-grained double covariance operator using multi-scale Itô calculus and sliding-window averaging. The construction explicitly incorporates two separated temporal scales: a fast microscopic fluctuation scale governing subquantum stochastic processes and a slower macroscopic observation scale associated with coarse-grained dynamics. Within the hydrodynamic limit, where the ratio between microscopic correlation time and averaging-window scale vanishes, rapidly fluctuating corrections disappear and the effective dynamics converges to a deterministic macroscopic transport equation. We show that the emergent macroscopic dynamics has the exact Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) form: coherent Hamiltonian evolution arises from deterministic subquantum flow, while dissipative channels emerge from quadratic noise correlations. The framework further demonstrates how non-separable interaction Hamiltonians can arise from strictly local, state-dependent stochastic feedback fields. In the fluctuation-free limit, the model reduces naturally to the standard von Neumann equation, providing a unified stochastic foundation for both open and closed quantum dynamics.