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Bridges between quantum and classical mechanics: Directed polymers, flocking and transitionless quanum driving

Loewe Yanez, Benjamin Andres

Published Apr 7, 2017• 0 citations
OpenAlex

Abstract

Often considered separated worlds, classical and quantum mechanics share numerous connections with one another. Indeed, as classical mechanics corresponds to a limiting case of quantum mechanics, certain concepts and elements of physical intuition developed in one theory can be and have been used to tackle issues in the other. Nevertheless, these connections do not only cover conceptual issues but also numerous techniques. Indeed, the inherently probabilistic nature of quantum mechanics and its close resemblance to Markovian stochastic processes opens the door to the application of a broad range of powerful methods, initially developed for quantum mechanics, in classical equilibrium and non-equilibrium statistical mechanics. In this thesis we develop progress in three subjects by taking advantage of either a conceptual or a methodological connection between quantum and classical mechanics. First, we develop the well-known mapping between systems of strongly repelling, two-dimensional directed lines and systems of one-dimensional fermions in order to extend the directed polymer problem to richer geometries. By expressing path integrals in generalized curvilinear coordinates, we successfully generalize the model to settings such as polymers anchored to curved edges, polymers constrained to uneven walls, and polymers constrained to curved surfaces. In each case, we identify the Hamiltonian of an analogous quantum system, which, because of the new geometry of each setting, develops features such as a time- and position-dependent mass and an external electromagnetic vector potential. Along the way, we perform an in-depth analysis of the approximations made and establish regimes of their validity. Finally, in order to obtain analytical results, we employ an extension of the time-dependent perturbation theory scheme of quantum mechanics to imaginary time. Complementing this with the assumption of ground-state dominance, we obtain compact expressions for universal shifts in the free energies of the various systems that allows for isolation of the effects of the distinct geometrical properties of each system. The second piece of work presented into this thesis relates to the non-equilibrium setting of active particles. Strongly interacting, self-propelled particles can form a spontaneously flowing, polar (i.e. motionally aligned), active fluid. The study of the connection between the microscopic dynamics of a single, self-propelled particle and the macroscopic dynamics of a liquid comprising such particles can yield insights into experimentally realizable active flows, but this connection is well understood in only a few select cases. Here, we introduce a model of self-propelled particles that is based on an analogy with the motion of an electron subject to strong spin-orbit coupling. We find that, within our model, self-propelled particles experience an analog of the Heisenberg uncertainly principle that instead relates positional and rotational noise. An extension to many-component (and hence more classical) spinors, under which this uncertainty relation vanishes, contributes to the justification of this interpretation. Furthermore, by coarse-graining the microscopic model, we find expressions for the coefficients of the hydrodynamic Toner-Tu equations, established some time ago to describe an active liquid composed of “active spins.” The connection between self-propelled particles and quantum spins may possibly provide a route for realizing exotic phases of matter using active liquids, via inspiration hailing from systems composed of strongly correlated electrons. The third and final piece of work to be presented on this thesis consists of a semi-classical approach to Transitionless Quantum Driving (TQD). TQD is a method, developed by means of a reverse engineering strategy, under which non-adiabatic transitions in time-dependent quantum systems are, in M. V. Berry's words, “stifled” through the introduction of a specific auxiliary Hamiltonian. This Hamiltonian comes, however, expressed as a formal sum of outer products of the original instantaneous eigenstates and their time-derivatives. Generically, how to actually create such an operator in the laboratory is thus rarely evident. The operator may even be non-local. By following a semi-classical approach, we obtain a recipe that yields the auxiliary Hamiltonian explicitly, in terms of the fundamental operators of the system (e.g., position and momentum). By using this formalism, we are able to ascertain criteria for the locality of the auxiliary Hamiltonian, and also to determine its exact form in certain special cases. Lastly, the explicit connection between the auxiliary Hamiltonian and the observables of the system allows for a perturbation scheme in cases in which an exact solution is not easily achievable. This scheme shows that, even in situations in which an exact local auxiliary term cannot be achieved, under special circumstances it is possible to achieve an operator that is local and still approximately stifles non-adiabatic transitions.

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Time-dependent quantum Markovian dynamics: from microscopic models to relaxation theory

Giovanni Di Meglio

The theory of open quantum systems represents one of the major developments of quantum mechanics over the last 50 years. Its central object of study is constituted by quantum systems that cannot be regarded as fully isolated, but instead interact with additional degrees of freedom, giving rise to irreversible phenomena such as dissipation and decoherence. The tools provided by this framework are nowadays of paramount importance across many active areas of research, including quantum computation, thermodynamics, metrology and biology. In this thesis, we investigate three main topics within the field. The first contribution concerns the microscopic derivation of the evolution equations, socalled quantum master equations, for open systems which are also subject to external driving fields. Standard techniques developed for time-independent Hamiltonians encounter several difficulties in the time-dependent case, largely restricting their applicability to specific instances, such as periodic or adiabatic drivings. In contrast, a core result of the derivation presented in the thesis is that, when the environment is sufficiently Markovian (essentially, memoryless), the non-adiabatic contributions to the driving field can be neglected in a controlled fashion, yielding an accurate master equation with time-dependent generator, capable of correctly reproducing the open dynamics even for strong drivings. This result is further supported by benchmarks against numerically exact simulations. The second contribution addresses the long-time behavior of general types of (Markovian) quantum master equations, including those of the form introduced above. While this problem has been extensively analyzed in the time-independent setting, virtually no results are known for time-dependent generators. In particular, we establish the foundations of a general theory of relaxation, that is, convergence towards an asymptotic trajectory in the long-time limit, independently of the initial state. The main results include a theorem guaranteeing sufficient conditions for relaxation and the derivation of a rate function governing the convergence to the asymptotic trajectory. The analysis is further extended to a simple non-Markovian case. Moreover, we treat in detail the specific master equation derived in the first part, connecting the aforementioned relaxation property directly to the underlying microscopic dynamics. The third contribution investigates chirality-induced spin selectivity (CISS) effects from the perspective of open quantum systems. Although the majority of theoretical studies aiming to model CISS are based on a coherent description of the spin dynamics, such effects have been observed also for electrons subject to incoherent transport across molecular networks. In this regard, we provide a rigorous ab initio derivation of the microscopic Hamiltonian describing the electronic transport, incorporating electron spin effects and contributions stemming from the nuclear vibrations. Within this result, we critically revise a master equation approach aiming to explain the origin of CISS effects observed in photosynthetic complexes. Taken together, these contributions offer fresh insights and original results on the interplay between driving fields and dissipation, relaxation phenomena, and electronic spin transport in the presence of incoherent dynamics. More broadly, these results provide new theoretical tools with several potential applications in quantum technologies, from modeling of control pulses in noisy environments to quantum reservoir engineering, and contribute to a deeper understanding of the role of quantum effects in biological systems.

•Jan 1, 2026•0 citations

Exploiting the isomorphism between quantum theory and classical statistical mechanics of polyatomic fluids

David Chandler, Peter Guy Wolynes

From a discretization of the path integral formulation of quantum mechanics, it is possible to relate equilibrium quantum many body theory to classical statistical mechanics. In this paper, we significantly extend and analyze this well known isomorphism in terms of the equilibrium theory of classical molecular fluids composed of flexible polyatomic species. We show how quantum influence functionals are isomorphic to classical cavity distribution functions. The former describe the influence of surrounding media on the dynamics of quantal degrees of freedom, and the latter describe environmental effects for classical models of flexible molecules and chemical equilibria. The connection allows the use of classical theories to perform nonperturbative calculations of influence functionals which treat the influence functionals and many body correlation functions in a self-consistent fashion. We illustrate the computational advantages of the method by studying its predictions for a hard sphere model of liquid helium above the l transition. The nature of quantum indistinguishability of identical particles (i.e., quantum exchange) is treated in our theory in terms of an exact isomorphism with chemical equilibria. This connection allows the treatment of exchange in condensed phases in terms of the classical law of mass action, and provides a computational advance over existing methods for interacting systems. By picturing exchange in terms of classical association equilibrium, we arrive at a view (hinted at long ago by Feynman and by Penrose and Onsager) in which the Bose condensation is related to an equilibrium polymeric sol–gel transition. Thus, below the l transition, the correlations in liquid helium are equivalent to those in a classical fluid containing a finite concentration of macroscopic polymers. We stress how the path integral aspect of the isomorphism leads to useful geometrical interpretations of quantum phenomena. For example, tunneling phenomena can be viewed in terms of solitonic (or instantonic) configurations or flexible chain molecules. With this picture, we show how the isomorphism can be employed to understand both adiabatic and nonadiabatic solvent effects on chemical bonding. For concreteness, we provide a detailed analysis of a particular model of the chemical bond for which a partial summation over intermediate quantum paths leads to an Ising model problem in the isomorphism. While applications like this are presented in the form of qualitative illustrations, a variety of methods can be employed to produce quantitative results. We sketch how these calculations can be performed for various problems, making connections with methods like the renormalization group (RG) technique. The classical isomorphism together with the modern theory of classical polyatomic systems provides a powerful framework for quantitative solutions of condensed matter quantum mechanical problems.

The Journal of Chemical Physics•Apr 1, 1981•1,308 citations

Nonequilibrium Quantum Systems: Fluctuations and Interactions

Yiğit Subaşı

We explore some aspects of nonequilibrium statistical mechanics of classical and quantum systems. Two chapters are devoted to fluctuation theorems which were originally derived for classical systems. The main challenge in formulating them in quantum mechanics is the fact that fundamental quantities of interest, like work, are defined via the classical concept of a phase space trajectory. We utilize the decoherent histories conceptual framework, in which classical trajectories emerge in quantum mechanics as a result of coarse graining, and provide a first-principles analysis of the nonequilibrium work relation of Jarzynski and Crooks's fluctuation theorem for a quantum system interacting with a general environment based on the quantum Brownian motion (QBM) model. We indicate a parameter range at low temperatures where the theorems might fail in their original form. Fluctuation theorems of Jarzynski and Crooks for systems obeying classical Hamiltonian dynamics are derived under the assumption that the initial conditions are sampled from a canonical ensemble, even though the equilibrium state of an isolated system is typically associated with the microcanonical ensemble. We address this issue through an exact analysis of the classical Brownian motion model. We argue that a stronger form of ensemble equivalence than usually discussed in equilibrium statistical mechanics is required for these theorems to hold in the infinite environment limit irrespective of the ensemble used, and proceed to prove it for this model. An exact expression for the probability distribution of work is obtained for finite environments. Intuitively one expects a system to relax to an equilibrium state when brought into contact with a thermal environment. Yet it is important to have rigorous results that provide conditions for equilibration and characterize the equilibrium state. We consider the dynamics of open quantum systems using the Langevin and master equations and rigorously show that under fairly general conditions quantum systems interacting with a heat bath relax to the equilibrium state defined as the reduced thermal state of the system plus environment, even in the strong coupling regime. Our proof is valid to second-order in interaction strength for general systems and exact for the linear QBM model, for which we also show the equivalence of multi-time correlations. In the final chapter we give a sampling of our investigations into macroscopic quantum phenomena. We work out in detail a specific example of how and under what conditions the center of mass (CoM) coordinate of a macroscopic object emerges as the relevant degree of freedom. Interaction patterns are studied in terms of the couplings they induce between the CoM and relative coordinates of two macroscopic objects. We discuss the implications of these interaction patterns on macroscopic entanglement.

•Jan 1, 2013•0 citations

Nonequilibrium generating functional and dynamics of coarse-grained variables

S. Majaniemi

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. 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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.

•May 14, 2004•1 citation

Directed-polymer systems explored via their quantum analogs

David Zeb Rocklin

The equilibrium statistical mechanics of classical directed polymers in D+1 dimensions is well known to be equivalent to the imaginary-time quantum dynamics of a quantum many-particle system in D spatial dimensions, with polymer con gurations corresponding to particle world-lines. This equivalence motivates the application of techniques originally designed for one-dimensional many-particle quantum systems to the exploration of many-polymer systems, as first recognized and exploited by P.-G. de Gennes [J. Chem. Phys. 48, 2257 (1968)]. In two dimensions, interactions give rise to an emergent polymer fluid, and I shall examine how topological constraints on this polymer fluid (e.g., due to uncrossable pins or barriers) and their geometry give rise to strong, entropy-driven forces. I shall also apply quantum techniques such as Bethe's Ansatz and bosonization to shed light on the structure of the polymer system. These techniques allow us to examine how polymer system correlations, thermodynamic properties, and response to impurities are influenced by strong polymer-polymer interactions. In three dimensions, polymers wind around one another and polymer topology may be incorporated via coupling to a Chern-Simons fi eld. As I discuss, this approach reveals the somewhat reduced role played by interactions in this higher-dimensional setting, leading to qualitatively diff erent polymer correlations, thermodynamic properties, and response to impurities.

•Aug 22, 2013•0 citations

Directed-polymer systems explored via their quantum analogs: Topological constraints and their consequences

D. Zeb Rocklin, Shina Tan, Paul M. Goldbart

The equilibrium statistical mechanics of classical directed polymers in two dimensions is well known to be equivalent to the imaginary-time quantum dynamics of a 1+1-dimensional many-particle system, with polymer configurations corresponding to particle world-lines. This equivalence motivates the application of techniques originally designed for one-dimensional many-particle quantum systems to the exploration of many-polymer systems, as first recognized and exploited by de Gennes [J. Chem. Phys. 48, 2257 (1968)]. In this low-dimensional setting interactions give rise to an emergent polymer fluid, and we examine how topological constraints on this polymer fluid (e.g., due to uncrossable pins or barriers) and their geometry give rise to strong, entropy-driven forces. In the limit of large polymer densities, in which a type of mean-field theory is accurate, we find that a pointlike pin causes a divergent pileup of polymer density on the high-density side of the pin and a zero-density region (or gap) of finite area on the low-density side. In addition, we find that the force acting on a pin that is only mildly displaced from its equilibrium position is sub-Hookean, growing less than linearly with the displacement, and that the gap created by the pin also grows sublinearly with the displacement. By contrast, the forces acting between multiple pins separated along the direction preferred by the polymers are super-Hookean. These nonlinear responses result from effective long-ranged interactions between polymer segments, which emerge via short-ranged interactions between distant segments of long polymer strands. In the present paper, we focus on the case of an infinitely strong, repulsive contact interaction, which ensures that the polymers completely avoid one another. In a companion paper, we consider the effects of a wider set of interpolymer interactions.

arXiv:1205.5027•Physical Review B•Oct 11, 2012•7 citations
Often considered separated worlds, classical and quantum mechanics share numerous connections with one another. Indeed, as classical mechanics corresponds to a limiting case of quantum mechanics, certain concepts and elements of physical intuition developed in one theory can be and have been used to tackle issues in the other. Nevertheless, these connections do not only cover conceptual issues but also numerous techniques. Indeed, the inherently probabilistic nature of quantum mechanics and its close resemblance to Markovian stochastic processes opens the door to the application of a broad range of powerful methods, initially developed for quantum mechanics, in classical equilibrium and non-equilibrium statistical mechanics. In this thesis we develop progress in three subjects by taking advantage of either a conceptual or a methodological connection between quantum and classical mechanics. First, we develop the well-known mapping between systems of strongly repelling, two-dimensional directed lines and systems of one-dimensional fermions in order to extend the directed polymer problem to richer geometries. By expressing path integrals in generalized curvilinear coordinates, we successfully generalize the model to settings such as polymers anchored to curved edges, polymers constrained to uneven walls, and polymers constrained to curved surfaces. In each case, we identify the Hamiltonian of an analogous quantum system, which, because of the new geometry of each setting, develops features such as a time- and position-dependent mass and an external electromagnetic vector potential. Along the way, we perform an in-depth analysis of the approximations made and establish regimes of their validity. Finally, in order to obtain analytical results, we employ an extension of the time-dependent perturbation theory scheme of quantum mechanics to imaginary time. Complementing this with the assumption of ground-state dominance, we obtain compact expressions for universal shifts in the free energies of the various systems that allows for isolation of the effects of the distinct geometrical properties of each system. The second piece of work presented into this thesis relates to the non-equilibrium setting of active particles. Strongly interacting, self-propelled particles can form a spontaneously flowing, polar (i.e. motionally aligned), active fluid. The study of the connection between the microscopic dynamics of a single, self-propelled particle and the macroscopic dynamics of a liquid comprising such particles can yield insights into experimentally realizable active flows, but this connection is well understood in only a few select cases. Here, we introduce a model of self-propelled particles that is based on an analogy with the motion of an electron subject to strong spin-orbit coupling. We find that, within our model, self-propelled particles experience an analog of the Heisenberg uncertainly principle that instead relates positional and rotational noise. An extension to many-component (and hence more classical) spinors, under which this uncertainty relation vanishes, contributes to the justification of this interpretation. Furthermore, by coarse-graining the microscopic model, we find expressions for the coefficients of the hydrodynamic Toner-Tu equations, established some time ago to describe an active liquid composed of “active spins.” The connection between self-propelled particles and quantum spins may possibly provide a route for realizing exotic phases of matter using active liquids, via inspiration hailing from systems composed of strongly correlated electrons. The third and final piece of work to be presented on this thesis consists of a semi-classical approach to Transitionless Quantum Driving (TQD). TQD is a method, developed by means of a reverse engineering strategy, under which non-adiabatic transitions in time-dependent quantum systems are, in M. V. Berry's words, “stifled” through the introduction of a specific auxiliary Hamiltonian. This Hamiltonian comes, however, expressed as a formal sum of outer products of the original instantaneous eigenstates and their time-derivatives. Generically, how to actually create such an operator in the laboratory is thus rarely evident. The operator may even be non-local. By following a semi-classical approach, we obtain a recipe that yields the auxiliary Hamiltonian explicitly, in terms of the fundamental operators of the system (e.g., position and momentum). By using this formalism, we are able to ascertain criteria for the locality of the auxiliary Hamiltonian, and also to determine its exact form in certain special cases. Lastly, the explicit connection between the auxiliary Hamiltonian and the observables of the system allows for a perturbation scheme in cases in which an exact solution is not easily achievable. This scheme shows that, even in situations in which an exact local auxiliary term cannot be achieved, under special circumstances it is possible to achieve an operator that is local and still approximately stifles non-adiabatic transitions.
The theory of open quantum systems represents one of the major developments of quantum mechanics over the last 50 years. Its central object of study is constituted by quantum systems that cannot be regarded as fully isolated, but instead interact with additional degrees of freedom, giving rise to irreversible phenomena such as dissipation and decoherence. The tools provided by this framework are nowadays of paramount importance across many active areas of research, including quantum computation, thermodynamics, metrology and biology. In this thesis, we investigate three main topics within the field. The first contribution concerns the microscopic derivation of the evolution equations, socalled quantum master equations, for open systems which are also subject to external driving fields. Standard techniques developed for time-independent Hamiltonians encounter several difficulties in the time-dependent case, largely restricting their applicability to specific instances, such as periodic or adiabatic drivings. In contrast, a core result of the derivation presented in the thesis is that, when the environment is sufficiently Markovian (essentially, memoryless), the non-adiabatic contributions to the driving field can be neglected in a controlled fashion, yielding an accurate master equation with time-dependent generator, capable of correctly reproducing the open dynamics even for strong drivings. This result is further supported by benchmarks against numerically exact simulations. The second contribution addresses the long-time behavior of general types of (Markovian) quantum master equations, including those of the form introduced above. While this problem has been extensively analyzed in the time-independent setting, virtually no results are known for time-dependent generators. In particular, we establish the foundations of a general theory of relaxation, that is, convergence towards an asymptotic trajectory in the long-time limit, independently of the initial state. The main results include a theorem guaranteeing sufficient conditions for relaxation and the derivation of a rate function governing the convergence to the asymptotic trajectory. The analysis is further extended to a simple non-Markovian case. Moreover, we treat in detail the specific master equation derived in the first part, connecting the aforementioned relaxation property directly to the underlying microscopic dynamics. The third contribution investigates chirality-induced spin selectivity (CISS) effects from the perspective of open quantum systems. Although the majority of theoretical studies aiming to model CISS are based on a coherent description of the spin dynamics, such effects have been observed also for electrons subject to incoherent transport across molecular networks. In this regard, we provide a rigorous ab initio derivation of the microscopic Hamiltonian describing the electronic transport, incorporating electron spin effects and contributions stemming from the nuclear vibrations. Within this result, we critically revise a master equation approach aiming to explain the origin of CISS effects observed in photosynthetic complexes. Taken together, these contributions offer fresh insights and original results on the interplay between driving fields and dissipation, relaxation phenomena, and electronic spin transport in the presence of incoherent dynamics. More broadly, these results provide new theoretical tools with several potential applications in quantum technologies, from modeling of control pulses in noisy environments to quantum reservoir engineering, and contribute to a deeper understanding of the role of quantum effects in biological systems.
From a discretization of the path integral formulation of quantum mechanics, it is possible to relate equilibrium quantum many body theory to classical statistical mechanics. In this paper, we significantly extend and analyze this well known isomorphism in terms of the equilibrium theory of classical molecular fluids composed of flexible polyatomic species. We show how quantum influence functionals are isomorphic to classical cavity distribution functions. The former describe the influence of surrounding media on the dynamics of quantal degrees of freedom, and the latter describe environmental effects for classical models of flexible molecules and chemical equilibria. The connection allows the use of classical theories to perform nonperturbative calculations of influence functionals which treat the influence functionals and many body correlation functions in a self-consistent fashion. We illustrate the computational advantages of the method by studying its predictions for a hard sphere model of liquid helium above the l transition. The nature of quantum indistinguishability of identical particles (i.e., quantum exchange) is treated in our theory in terms of an exact isomorphism with chemical equilibria. This connection allows the treatment of exchange in condensed phases in terms of the classical law of mass action, and provides a computational advance over existing methods for interacting systems. By picturing exchange in terms of classical association equilibrium, we arrive at a view (hinted at long ago by Feynman and by Penrose and Onsager) in which the Bose condensation is related to an equilibrium polymeric sol–gel transition. Thus, below the l transition, the correlations in liquid helium are equivalent to those in a classical fluid containing a finite concentration of macroscopic polymers. We stress how the path integral aspect of the isomorphism leads to useful geometrical interpretations of quantum phenomena. For example, tunneling phenomena can be viewed in terms of solitonic (or instantonic) configurations or flexible chain molecules. With this picture, we show how the isomorphism can be employed to understand both adiabatic and nonadiabatic solvent effects on chemical bonding. For concreteness, we provide a detailed analysis of a particular model of the chemical bond for which a partial summation over intermediate quantum paths leads to an Ising model problem in the isomorphism. While applications like this are presented in the form of qualitative illustrations, a variety of methods can be employed to produce quantitative results. We sketch how these calculations can be performed for various problems, making connections with methods like the renormalization group (RG) technique. The classical isomorphism together with the modern theory of classical polyatomic systems provides a powerful framework for quantitative solutions of condensed matter quantum mechanical problems.
We explore some aspects of nonequilibrium statistical mechanics of classical and quantum systems. Two chapters are devoted to fluctuation theorems which were originally derived for classical systems. The main challenge in formulating them in quantum mechanics is the fact that fundamental quantities of interest, like work, are defined via the classical concept of a phase space trajectory. We utilize the decoherent histories conceptual framework, in which classical trajectories emerge in quantum mechanics as a result of coarse graining, and provide a first-principles analysis of the nonequilibrium work relation of Jarzynski and Crooks's fluctuation theorem for a quantum system interacting with a general environment based on the quantum Brownian motion (QBM) model. We indicate a parameter range at low temperatures where the theorems might fail in their original form. Fluctuation theorems of Jarzynski and Crooks for systems obeying classical Hamiltonian dynamics are derived under the assumption that the initial conditions are sampled from a canonical ensemble, even though the equilibrium state of an isolated system is typically associated with the microcanonical ensemble. We address this issue through an exact analysis of the classical Brownian motion model. We argue that a stronger form of ensemble equivalence than usually discussed in equilibrium statistical mechanics is required for these theorems to hold in the infinite environment limit irrespective of the ensemble used, and proceed to prove it for this model. An exact expression for the probability distribution of work is obtained for finite environments. Intuitively one expects a system to relax to an equilibrium state when brought into contact with a thermal environment. Yet it is important to have rigorous results that provide conditions for equilibration and characterize the equilibrium state. We consider the dynamics of open quantum systems using the Langevin and master equations and rigorously show that under fairly general conditions quantum systems interacting with a heat bath relax to the equilibrium state defined as the reduced thermal state of the system plus environment, even in the strong coupling regime. Our proof is valid to second-order in interaction strength for general systems and exact for the linear QBM model, for which we also show the equivalence of multi-time correlations. In the final chapter we give a sampling of our investigations into macroscopic quantum phenomena. We work out in detail a specific example of how and under what conditions the center of mass (CoM) coordinate of a macroscopic object emerges as the relevant degree of freedom. Interaction patterns are studied in terms of the couplings they induce between the CoM and relative coordinates of two macroscopic objects. We discuss the implications of these interaction patterns on macroscopic entanglement.
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.
The equilibrium statistical mechanics of classical directed polymers in D+1 dimensions is well known to be equivalent to the imaginary-time quantum dynamics of a quantum many-particle system in D spatial dimensions, with polymer con gurations corresponding to particle world-lines. This equivalence motivates the application of techniques originally designed for one-dimensional many-particle quantum systems to the exploration of many-polymer systems, as first recognized and exploited by P.-G. de Gennes [J. Chem. Phys. 48, 2257 (1968)]. In two dimensions, interactions give rise to an emergent polymer fluid, and I shall examine how topological constraints on this polymer fluid (e.g., due to uncrossable pins or barriers) and their geometry give rise to strong, entropy-driven forces. I shall also apply quantum techniques such as Bethe's Ansatz and bosonization to shed light on the structure of the polymer system. These techniques allow us to examine how polymer system correlations, thermodynamic properties, and response to impurities are influenced by strong polymer-polymer interactions. In three dimensions, polymers wind around one another and polymer topology may be incorporated via coupling to a Chern-Simons fi eld. As I discuss, this approach reveals the somewhat reduced role played by interactions in this higher-dimensional setting, leading to qualitatively diff erent polymer correlations, thermodynamic properties, and response to impurities.
The equilibrium statistical mechanics of classical directed polymers in two dimensions is well known to be equivalent to the imaginary-time quantum dynamics of a 1+1-dimensional many-particle system, with polymer configurations corresponding to particle world-lines. This equivalence motivates the application of techniques originally designed for one-dimensional many-particle quantum systems to the exploration of many-polymer systems, as first recognized and exploited by de Gennes [J. Chem. Phys. 48, 2257 (1968)]. In this low-dimensional setting interactions give rise to an emergent polymer fluid, and we examine how topological constraints on this polymer fluid (e.g., due to uncrossable pins or barriers) and their geometry give rise to strong, entropy-driven forces. In the limit of large polymer densities, in which a type of mean-field theory is accurate, we find that a pointlike pin causes a divergent pileup of polymer density on the high-density side of the pin and a zero-density region (or gap) of finite area on the low-density side. In addition, we find that the force acting on a pin that is only mildly displaced from its equilibrium position is sub-Hookean, growing less than linearly with the displacement, and that the gap created by the pin also grows sublinearly with the displacement. By contrast, the forces acting between multiple pins separated along the direction preferred by the polymers are super-Hookean. These nonlinear responses result from effective long-ranged interactions between polymer segments, which emerge via short-ranged interactions between distant segments of long polymer strands. In the present paper, we focus on the case of an infinitely strong, repulsive contact interaction, which ensures that the polymers completely avoid one another. In a companion paper, we consider the effects of a wider set of interpolymer interactions.