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Energy Distribution: Reconstructing the First Principles Framework of Quantum Theory Universal Formula for Internal Dissipation Power in Many Body Systems and Rigorous Derivation of the Mesoscopic Diffusion Equation

Ende Guo

Published Feb 27, 2026• 0 citations
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Abstract

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 ($Q$), an interfacial defect network ($D$), and bulk channels ($B$). 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 $Q$ and $D$, and deriving the universal formula for internal dissipation power $R_{\mathrm{int}} = \hbar\omega_q[\Gamma_\downarrow\langle n\rangle - \Gamma_\uparrow(1-\langle n\rangle)]$, where the relaxation rates are completely determined by the internal structure of $D$, revealing the core role of the energy distribution ratio $f$. This paper generalizes this new paradigm based on energy distribution from single-body to many-body systems. The core step is to define multiple $Q$ contacts and their shared $D$ and $B$ 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 $D$ network. Through rigorous coarse-graining of this overall system, this paper derives the governing equation for the evolution of its internal energy density field $u(\mathbf{r},t)$—a classical reaction-diffusion equation:\[\frac{\partial u}{\partial t} = R_{\mathrm{total}} - \gamma_0(\mathbf{r}) u + \nabla \cdot (\gamma_2 \nabla u).\]Here, the local dissipation rate $\gamma_0(\mathbf{r})$ and the diffusion coefficient $\gamma_2$ are completely determined by the properties of the $D$ network, corresponding respectively to the spatial distribution of the energy distribution ratio $f$ 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 \(f\) 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 \(\Gamma_{ij}\) 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 \(u(\mathbf{r},t)\) and the environmental field energy density \(U(\mathbf{r},t)\): \[\frac{\partial u}{\partial t} = R_{\mathrm{total}}(\mathbf{r},t) - \gamma_0(U) u + \nabla \cdot \bigl( \gamma_2(U) \nabla u \bigr)\]\[\frac{\partial U}{\partial t} = D \nabla^2 U - \Gamma_B U + \rho(\mathbf{r}) \gamma_0(U) u\] Within this framework, the energy distribution ratio \(f(\mathbf{r},t) = \gamma_0 u / R_{\mathrm{total}}\) emerges naturally as a macroscopic quantity, its physical essence being jointly determined by the system’s microscopic coupling structure (embodied by \(\Gamma_{ij}\)) 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.

DOI
10.5281/zenodo.18796157

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•May 14, 2004•1 citation
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:[\frac{\partial u}{\partial t} = R_{\mathrm{total}} - \gamma_0(\mathbf{r}) u + \nabla \cdot (\gamma_2 \nabla 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 (f) 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 (\Gamma_{ij}) 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(\mathbf{r},t)) and the environmental field energy density (U(\mathbf{r},t)): [\frac{\partial u}{\partial t} = R_{\mathrm{total}}(\mathbf{r},t) - \gamma_0(U) u + \nabla \cdot \bigl( \gamma_2(U) \nabla u \bigr)][\frac{\partial U}{\partial t} = D \nabla^2 U - \Gamma_B U + \rho(\mathbf{r}) \gamma_0(U) u] Within this framework, the energy distribution ratio (f(\mathbf{r},t) = \gamma_0 u / R_{\mathrm{total}}) emerges naturally as a macroscopic quantity, its physical essence being jointly determined by the system’s microscopic coupling structure (embodied by (\Gamma_{ij})) 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 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
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.
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
We present a general and practical theoretical framework to investigate how energy is dissipated in open quantum system dynamics. This is performed by quantifying the contributions of individual bath components to the overall dissipation of the system. The framework is based on the Nakajima-Zwanzig projection operator technique, which allows us to express the rate of energy dissipation into a specific bath degree of freedom by using traces of operator products. The approach captures system-bath interactions to all orders, but is based on second-order perturbation theory on the off-diagonal subsystem's couplings and a Markovian description of the bath. The usefulness of our theory is demonstrated by applying it to various models of open quantum systems involving harmonic oscillators or spin baths and connecting the outcomes to existing results such as our previously reported formula derived for locally coupled harmonic baths [Kim and Franco, J. Chem. Phys. 154, 084109 (2021)]. We also prove that the dissipation calculated by our theory rigorously satisfies thermodynamic principles such as energy conservation and detailed balance. Overall, the strategy can be used to develop the theory and simulation of dissipation pathways to interpret and engineer the dynamics of open quantum systems.
Update Notice (v2) 2026/2/13 This preprint has been superseded by a significantly revised version, which has been uploaded as a separate submission. In the original version (v1), the relation between internal dissipation power and the energy relaxation rate was introduced in a phenomenological manner. However, a rigorous first-principles derivation has now been completed in a companion paper [Guo (2026), Exact Relation between Internal Dissipation Power and Energy Relaxation Rate——A Rigorous Derivation in Markovian Quantum Systems]. Using this exact relation Rinternal=ℏωqΓ1⟨σ+σ−⟩Rinternal=ℏωqΓ1⟨σ+σ−⟩, the previously phenomenological "energy competition" framework has been reformulated from the ground up. As a result: The parameter ff (energy allocation fraction) is now rigorously defined in terms of experimentally accessible quantities, rather than introduced as a phenomenological fit parameter. The previous phenomenological arguments and associated experimental discussions in v1 are no longer valid and should be disregarded. The new version provides a complete microscopic foundation for the theory of energy competition in superconducting qubit decoherence. Readers are strongly encouraged to consult the latest version (available at [10.5281/zenodo.18624328) for a comprehensive and self-contained treatment. Version 2 The decoherence time of superconducting qubits is commonly believed to be dominatedby dielectric loss (loss tangent tan 𝛿). Multiple experimental studies have found an empiricalscaling relationship approximately Γ ∝ (tan 𝛿)0.7 between the decoherence rate Γ = 1/𝑇2and tan 𝛿. Based on the ”Interface Projection Realism” framework, this paper proposes apossible new physical picture for decoherence. By analyzing 25 superconducting quantumdevices covering single-qubit, multi-qubit, hybrid systems, and 3D integration, we observethe dispersion of this scaling relationship in optimized interface samples. Experimentaldata show that unoptimized samples indeed follow Γ ∝ (tan 𝛿)0.7, while optimized samplesexhibit significant dispersion in the scaling exponent.To explain this phenomenon, we propose a decoherence theoretical framework basedon the ”Energy Competition” picture. This framework reinterprets decoherence as the re-sult of competitive allocation of the input microwave energy between the ”internal dissipa-tion channel generating noise” and the ”net channel maintaining coherence.” This physicalpicture is a concrete implementation of Interface Projection Realism in the decoherenceproblem. By introducing the energy allocation ratio 𝑓 and the interface intrinsic competi-tion coefficient 𝜒, the theoretical framework predicts that the pure dephasing rate satisfiesΓ𝜙 = 𝜒 · 𝑓 /(1 − 𝑓 ) in the strong suppression limit. This framework provides a possibleexplanation for the dispersion of scaling behavior. Furthermore, the analysis of paired sam-ples (with the same tan 𝛿 but 𝑇2 differing by a factor of 16.7) provides indirect evidence forthe possibility of 𝑓 being an independent parameter.Moreover, the theoretical framework can unify the explanation of decoherence data fromsingle qubits to complex systems and proposes unique predictions that can be directly veri-fied experimentally, such as power independence and noise spectrum evolution. This workprovides a new perspective based on Interface Projection Realism for understanding quan-tum decoherence and deepens the goal of interface engineering from simply reducing lossto possibly regulating the energy allocation ratio 𝑓 . Important Update (Updated January 28, 2026) The square-root decoherence scaling law (Γ∝κeff) with scaling exponent α=0.5 reported in this work was derived via empirical fitting against superconducting qubit experimental datasets, based on the "resource competition theory at dynamic projection interfaces." This exponent was established under the framework of phenomenological analysis, leveraging established physical heuristics for intrinsic material loss and the loss angle tanδ, and exhibited high fitting goodness for low-noise, weak-coupling quantum systems. Subsequent research has advanced this framework to a comprehensive mathematical-physical theory—Interface Projection Realism (IPR)—grounded in quantum information first principles. A key innovation of IPR is the introduction of the transmissivity field Ξ(x,t) (as the fundamental order parameter describing the quantum-classical interface) and the rigorous derivation of its quantum information dynamics equation. By analyzing an expanded set of superconducting qubit decoherence data through weighted least-squares fitting (yielding R2=0.9987, an extremely significant F-statistic of 2856.3), the universal decoherence scaling exponent is refined to α=0.792±0.008. This revised exponent corresponds to a synergy index m=4.81±0.15, which originates from the cooperative effect of five major noise channels (dielectric loss, quasiparticle tunneling, flux noise, charge noise, and radiation loss) embedded in quantum channel dynamics. IPR further makes two independent testable predictions—equivalence principle violation (Δa/aˉ∼2.3×10−9) and a first-principles calculation of dark energy density (ρΛ≈4.2×10−47GeV4, consistent with Planck 2018 observations)—demonstrating its explanatory power across quantum decoherence, quantum gravity, and cosmology. A complete account of the theoretical derivation, microphysical mechanism, and experimental corroboration is provided in our latest work: Interface Projection Realism: A Multiscale Unified Theory Based on Quantum Information Dynamics (Zenodo, Version v3, DOI: 10.5281/zenodo.18400122, January 28, 2026). This preprint serves as a foundational stage-specific contribution to the evolutionary trajectory of the IPR framework. The α=0.5 square-root scaling relation remains a valid limiting case for low-noise, weak-coupling material systems, while the refined α=0.792 exponent offers a universal description across broader material and noise conditions. Version 1 We report a universal square-root scaling law between the decoherence rate Γ of a quantum system and its effective intrinsic dissipation κeff, i.e., Γ ∝ √ κeff. This empirical law, observed across platforms like superconducting qubits, finds its mechanistic explanation within a new theoretical framework of resource com petition at a dynamic projection interface. The framework posits that our (3+1)-dimensional classical spacetime emerges from a higher-dimensional realm of potentialities (L1) through a filtering process governed by the interface perme ability field Ξ(x, t). Finite information-energy flux from L1 is contested between systems’ tendency to maintain quantum coherence and their intrinsic demand to dissipate noise for classical stability, quantified by κeff. From this competition dy namics, we derive the square-root scaling law. The model further provides unified, objective descriptions of quantum measurement as channel locking and of grav ity as emergence from Ξ-field distortion. Several concrete, testable predictions are proposed, including nonlinear jumps in coherence at phase transitions and a novel testable correlation for equivalence principle experiments.
A generating functional formalism is developed to facilitate the derivation of coarse-grained dynamics of macroscopically relevant variables in various types of many-body problems. The relevant variables can be found either by using symmetry analysis of the action or experimental data. The outcome of the coarse-graining process is a set of coupled partial differential equations for the expectation values of the relevant observables. The set of the macroscopically important observables consists typically of conserved variables and order operators, which are related to the broken symmetries of the system. Symmetry breaking (spontaneous or explicit) in the formalism can be dynamically induced by coupling the system to a generalized heat bath.\n\nThe nonequilibrium generating functional formalism is applied to coarse-graining of generalized hydrodynamics of macroscopic quantum phenomena such as Bose fluids and superconductors, and finally classical fluids. The method is by no means restricted to fluids, but can be applied to condensed matter problems as well. High density systems are in general more problematic and it remains to be seen if one can come up with more powerful approximation methods within the current formalism in the future.\n\nThe generating functional formalism is shown to reproduce in the specific limits the well-known ground state, finite temperature and time-dependent density functional theories in the quantum regime. We enlarge the set of relevant variables to include also non-conserved order-operators and show how to form generalized density functional theories, which can be by construction linked to the phase-transition behaviour of the system. They are extensions of the traditional density functional theories that were not developed to describe phase changes but only facilitated the expression of the system's (free) energy in terms of the conserved density variable (e.g. number density of electrons).\n\nWe develop a generalized density functional theory (effective action of the generating functional) for superconductors and superfluids. For superconductors it is shown by simple symmetry analysis that the relevant order parameter is the pair field of two electrons. Coarse-graining out the phononic degrees of freedom is shown to lead to a retarded effective interaction between the electrons. In the local static limit (effective interaction between electrons static) the theory reduces to the time-dependent Ginzburg-Landau theory. Similarly, by symmetry arguments, the order parameter of a weakly interacting Bose fluid is identified and its dynamics in the tree level approximation is shown to follow the Gross-Pitaevskii equation. Higher order corrections can be systematically produced.\n\nClassical molecular dynamics and stochastic cellular automata types of higher level descriptions can be mapped into the same formalism using commonly known techniques. It is also shown how to connect the static and dynamic density functionals to phase-field models (models of critical dynamics), which are commonly used in materials science modelling. A specific example is provided for a liquid-gas-solid system. The phase-field modelling is then applied to the derivation of dynamics of the liquid-gas phase boundaries and triple lines for which a new projection operator technique is developed. We explain in detail how to go from microscopics to macroscopics and relate the effective parameters on each level to each other. For example, we express the macroscopic surface tensions from the phase-field model in terms of the molecular dynamics simulation parameters. We also discuss how to extract the relevant cross-over time and length scales from the theoretical description and relate them to the experimentally measurable ones. It is shown how the coupling of the triple line and the liquid-gas boundary leads to emergence of memory effects. For confined geometry we also derive a new form for the restoring force acting on the contact line. The phase-field model of the liquid-gas-solid system is first discussed without explicit hydrodynamics from the point of view of diffuse mass transfer and then hydrodynamic effects (momentum conservation) are added.\n\nEvolution equations are derived for the mass, momentum and energy densities representing the proper set of macro variables of simple fluids. The derivation is based on a macroscopic Poisson bracket formulation of phase-field dynamics. Navier-Stokes equations are obtained with extra force terms arising from wall-fluid interactions and capillary stresses. It is also shown how the hydrodynamics boundary conditions for stress tensor can be derived from the bulk fields alone in the sharp interface limit. Some mechanisms for generation of the no-slip boundary condition are also hypothesized. These results are important because they show how in general boundary conditions and corrections to them can be derived and how the numerical implementation can benefit from such a formulation of hydrodynamics, where all boundary conditions are expressed in terms of bulk fields alone. Again, these results have wider applicability than the setting where they have been derived.\n\nThe advantage of a systematic coarse-graining procedure enables us to clarify the role of the order parameter concept in the phase-field description of one and two component fluids. We also consider different ways of including gravitational effects and apply dimensional analysis to obtain a condition for the onset of capillary turbulence.\n\nFinally, we present some new formalism for stochastic processes, where the relevant field variables are embedded in an environment characterized by quenched disorder. Whenever some lower dimensional (e.g. defect) structures are projected out of the bulk field evolution equations containing either frozen of coarse-graining generated noise fields, a more complicated effective stochastic noise component will appear in the projected equation of motion of the defect. In particular, the effective noise can be long-ranged even if the quenched background disorder is not. Several condensed matter and hydrodynamical examples describing dynamics of interfaces and line-like objects such as polymers and domain wall defects are considered.