Motivated by the question of stability, in this letter we argue that an effective quantum-like theory can emerge in complex adaptive systems. In the concrete example of stochastic Lotka-Volterra dynamics, the relevant effective "Planck constant" associated with such emergent "quantum" theory has the dimensions of the square of the unit of time. Such an emergent quantum-like theory has inherently non-classical stability as well as coherent properties that are not, in principle, endangered by thermal fluctuations and therefore might be of crucial importance in complex adaptive systems.
Motivated by the question of stability, in this letter we argue that an effective quantum-like theory can emerge in complex adaptive systems. In the concrete example of stochastic Lotka-Volterra dynamics, the relevant effective "Planck constant" associated with such emergent "quantum" theory has the dimensions of the square of the unit of time. Such an emergent quantum-like theory has inherently non-classical stability as well as coherent properties that are not, in principle, endangered by thermal fluctuations and therefore might be of crucial importance in complex adaptive systems.
We explore the concept of emergent quantum-like theory in complex adaptive systems, and examine in particular the concrete example of such an emergent (or “mock”) quantum theory in the Lotka–Volterra system. In general, we investigate the possibility of implementing the mathematical formalism of quantum mechanics on classical systems, and what would be the conditions for using such an approach. We start from a standard description of a classical system via Hamilton–Jacobi (HJ) equation and reduce it to an effective Schrodinger-type equation, with a (mock) Planck constant , which is system-dependent. The condition for this is that the so-called quantum potential VQ, which is state-dependent, is cancelled out by some additional term in the HJ equation. We consider this additional term to provide for the coupling of the classical system under consideration to the ‘environment’. We assume that a classical system could cancel out the VQ term (at least approximately) by fine tuning to the environment. This might provide a mechanism for establishing a stable, stationary states in (complex) adaptive systems, such as biological systems. In particular, we present a general argument as to why the non-equilibrium dynamics of a classical system could lead to a mock quantum description that ensures stability compatible with adaptability. In this context we emphasize the state dependent nature of the mock quantum dynamics and we also introduce the new concept of the mock quantum, state dependent, statistical field theory. We also discuss some universal features of the quantum-to-classical as well as the mock-quantum-to-classical transition found in the turbulent phase of the hydrodynamic formulation of our proposal. In this way we re-frame the concept of decoherence into the concept of ‘quantum turbulence’, i.e. that the transition between quantum and classical could be defined in analogy to the transition from laminar to turbulent flow in hydrodynamics.
We explore the concept of emergent quantum-like theory in complex adaptive systems, and examine in particular the concrete example of such an emergent (or "mock") quantum theory in the Lotka-Volterra system. In general, we investigate the possibility of implementing the mathematical formalism of quantum mechanics on classical systems, and what would be the conditions for using such an approach. We start from a standard description of a classical system via Hamilton-Jacobi (HJ) equation and reduce it to an effective Schr\"odinger-type equation, with a (mock) Planck constant $\mockbar$, which is system-dependent. The condition for this is that the so-called quantum potential VQ, which is state-dependent, is cancelled out by some additional term in the HJ equation. We consider this additional term to provide for the coupling of the classical system under consideration to the "environment." We assume that a classical system could cancel out the VQ term (at least approximately) by fine tuning to the environment. This might provide a mechanism for establishing a stable, stationary states in (complex) adaptive systems, such as biological systems. In this context we emphasize the state dependent nature of the mock quantum dynamics and we also introduce the new concept of the mock quantum, state dependent, statistical field theory. We also discuss some universal features of the quantum-to-classical as well as the mock-quantum-to-classical transition found in the turbulent phase of the hydrodynamic formulation of our proposal. In this way we reframe the concept of decoherence into the concept of "quantum turbulence," i.e. that the transition between quantum and classical could be defined in analogy to the transition from laminar to turbulent flow in hydrodynamics.
We explore the concept of emergent quantum-like theory in complex adaptive systems, and examine in particular the concrete example of such an emergent (or "mock") quantum theory in the Lotka-Volterra system. In general, we investigate the possibility of implementing the mathematical formalism of quantum mechanics on classical systems, and what would be the conditions for using such an approach. We start from a standard description of a classical system via Hamilton-Jacobi (HJ) equation and reduce it to an effective Schrödinger-type equation, with a (mock) Planck constant $\mockbar$, which is system-dependent. The condition for this is that the so-called quantum potential VQ, which is state-dependent, is cancelled out by some additional term in the HJ equation. We consider this additional term to provide for the coupling of the classical system under consideration to the "environment." We assume that a classical system could cancel out the VQ term (at least approximately) by fine tuning to the environment. This might provide a mechanism for establishing a stable, stationary states in (complex) adaptive systems, such as biological systems. In this context we emphasize the state dependent nature of the mock quantum dynamics and we also introduce the new concept of the mock quantum, state dependent, statistical field theory. We also discuss some universal features of the quantum-to-classical as well as the mock-quantum-to-classical transition found in the turbulent phase of the hydrodynamic formulation of our proposal. In this way we reframe the concept of decoherence into the concept of "quantum turbulence," i.e. that the transition between quantum and classical could be defined in analogy to the transition from laminar to turbulent flow in hydrodynamics.
The question of how statistical physics can be seen to emerge from unitary quantum dynamics goes back to Schrödinger and von Neumann, however has been reaffirmed as a topic of importance due to recent advances in experimental capabilities, allowing the observation of the unitary evolution of many-particle systems over long periods of time. Understanding why such a system evolves to the specific thermal equilibrium state, effectively described by a relevant thermodynamic ensemble, has thus seen significantly more interest in recent years. Furthermore, observations of the dynamics of such systems open questions on the route to equilibrium, and subsequent fluctuations. In this thesis I will develop an approach to this problem, taking as a starting point the non-integrability of the system under study, which leads to a generic model in terms of random matrix theory (RMT), and more generally, chaotic wavefunctions. In the formulation of these methods, special attention is paid to the form of local observables of quantum systems, and an approach to their description is developed in terms of correlation functions of chaotic wavefunctions. From this approach a key conjecture in the understanding of thermalization, the eigenstate thermalization hypothesis (ETH), is derived in full, and the dynamical behaviour of such observables is obtained analytically. Further, I will show that emergent classical behaviour can be observed in the form of a fluctuation-dissipation theorem (FDT) of Brownian processes. This is exploited to develop an experimental proposal to measure the ‘complexity’ of a quantum device, by experimentally obtaining its effective Hilbert space dimension in terms of a fully connected system. Finally, quantum jump trajectories, describing stroboscopic projective measurements of local observables are studied, and shown to display classical Brownian dynamics in the form of a Markov process. Throughout this thesis, exact diagonalization calculations of realistic quantum spin systems are used to confirm analytical results.
Quantum mechanics is traditionally formulated in terms of postulates inferred from experimentalobservations. Here we propose that the statistical regularities of quantum mechanics arise as macro-scopic manifestations of stochastic fluctuations of dynamical parameters in microscopic systems gen-erated by a non-inertial physical background. Although the specific nature of this background mayvary, the universality of quantum statistics is ensured by invariant properties of stochastic dynam-ics - analogous to the role of the equivalence principle in general relativity - and by the existence ofa fundamental action scale, Planck constant h. By extending classical mechanics to include higher-order derivatives induced by non-inertial fuctuations and imposing a stability condition on admissibletrajectories, the minimal action scale h naturally emerges. In this framework quantum mechanicsappears as an effective statistical description of systems embedded in a fluctuating non-inertial envi-ronment while preserving all standard predictions of the theory.
Since Newton's time, deterministic causality has been considered a crucial prerequisite in any fundamental theory in physics. In contrast, the present work investigates stochastic dynamical models for motion in one spatial dimension, in which Newtonian mechanics becomes an emergent property: We present a coherent theory in which a Hamilton-Jacobi equation (HJE) emerges in a description of the evolution of entropy −ϕ(x,t)=εlog(Probability) of a system under observation and in the limit of large information extent ε−1 in homogeneous space and time. The variable ϕ represents a non-random high-order statistical concept that is distinct from probability itself as ε=0; the HJE embodies an emergent law of deterministic causality in continuous space and time with an Imaginary Scale symmetry (t,x,ϕ)↔(it,ix,−iϕ). ϕ(x,t) exhibits a nonlinear wave phenomenon with a mathematical singularity in finite time, overcoming which we introduce viscosity ε(∂2ϕ/∂x2) and wave iε(∂2ϕ/∂x2) perturbations, articulating dissipation and conservation, which break the Imaginary Scale symmetry: They lead to the Brownian motion and Schrödinger's equation of motion, respectively. Last but not least, Lagrange's action in classical mechanics acquires an entropic interpretation and Hamilton's principle is established.