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Hong Qian - One of the best experts on this subject based on the ideXlab platform.
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a thermodynamic theory of ecology Helmholtz Theorem for lotka volterra equation extended conservation law and stochastic predator prey dynamics
Proceedings of The Royal Society A: Mathematical Physical and Engineering Sciences, 2015Co-Authors: Hong QianAbstract:We carry out mathematical analyses, a la Helmholtz’s and Boltzmann’s 1884 studies of monocyclic Newtonian dynamics, for the Lotka–Volterra (LV) equation exhibiting predator–prey oscillations. In doing so, a novel ‘thermodynamic theory’ of ecology is introduced. An important feature, absent in the classical mechanics, of ecological systems is a natural stochastic population dynamic formulation of which the deterministic equation (e.g. the LV equation studied) is the infinite population limit. Invariant density for the stochastic dynamics plays a central role in the deterministic LV dynamics. We show how the conservation law along a single trajectory extends to incorporate both variations in a model parameter α and in initial conditions: Helmholtz’s Theorem establishes a broadly valid conservation law in a class of ecological dynamics. We analyse the relationships among mean ecological activeness θ , quantities characterizing dynamic ranges of populations A and α , and the ecological force F α . The analyses identify an entire orbit as a stationary ecology, and establish the notion of an ‘equation of ecological states’. Studies of the stochastic dynamics with finite populations show the LV equation as the robust, fast cyclic underlying behaviour. The mathematical narrative provides a novel way of capturing long-term dynamical behaviours with an emergent conservative ecology .
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A thermodynamic theory of ecology: Helmholtz Theorem for Lotka–Volterra equation, extended conservation law, and stochastic predator–prey dynamics
Proceedings of The Royal Society A: Mathematical Physical and Engineering Sciences, 2015Co-Authors: Yi-an Ma, Hong QianAbstract:We carry out mathematical analyses, a la Helmholtz’s and Boltzmann’s 1884 studies of monocyclic Newtonian dynamics, for the Lotka–Volterra (LV) equation exhibiting predator–prey oscillations. In doing so, a novel ‘thermodynamic theory’ of ecology is introduced. An important feature, absent in the classical mechanics, of ecological systems is a natural stochastic population dynamic formulation of which the deterministic equation (e.g. the LV equation studied) is the infinite population limit. Invariant density for the stochastic dynamics plays a central role in the deterministic LV dynamics. We show how the conservation law along a single trajectory extends to incorporate both variations in a model parameter α and in initial conditions: Helmholtz’s Theorem establishes a broadly valid conservation law in a class of ecological dynamics. We analyse the relationships among mean ecological activeness θ , quantities characterizing dynamic ranges of populations A and α , and the ecological force F α . The analyses identify an entire orbit as a stationary ecology, and establish the notion of an ‘equation of ecological states’. Studies of the stochastic dynamics with finite populations show the LV equation as the robust, fast cyclic underlying behaviour. The mathematical narrative provides a novel way of capturing long-term dynamical behaviours with an emergent conservative ecology .
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Universal ideal behavior and macroscopic work relation of linear irreversible stochastic thermodynamics
New Journal of Physics, 2015Co-Authors: Yi-an Ma, Hong QianAbstract:We revisit the Ornstein–Uhlenbeck (OU) process as the fundamental mathematical description of linear irreversible phenomena, with fluctuations, near an equilibrium. By identifying the underlying circulating dynamics in a stationary process as the natural generalization of classical conservative mechanics, a bridge between a family of OU processes with equilibrium fluctuations and thermodynamics is established through the celebrated Helmholtz Theorem. The Helmholtz Theorem provides an emergent macroscopic ‘equation of state’ of the entire system, which exhibits a universal ideal thermodynamic behavior. Fluctuating macroscopic quantities are studied from the stochastic thermodynamic point of view and a non-equilibrium work relation is obtained in the macroscopic picture, which may facilitate experimental study and application of the equalities due to Jarzynski, Crooks, and Hatano and Sasa.
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The Helmholtz Theorem for the Lotka-Volterra Equation, the Extended Conservation Relation, and Stochastic Predator-Prey Dynamics
arXiv: Mathematical Physics, 2014Co-Authors: Yi-an Ma, Hong QianAbstract:We carry out a mathematical analysis, \`{a} la Helmholtz's and Boltzmann's 1884 studies of monocyclic Newtonian mechanics, for the Lotka-Volterra (LV) equation exhibiting oscillatory predator-prey dynamics. One of the important features of the latter system, absent in the classical mechanical model, is a natural stochastic dynamic formulation of which the LV equation is the infinite population limit. The invariant density for the stochastic dynamics plays a central role in the deterministic LV dynamics. We show how the conservation law along a single trajectory can be extended to incorporate both variations in model parameter $\alpha$ and in the initial conditions: Helmholtz's Theorem establishes a broadly valid conservation law in a class of ecological dynamics. We analyze the relationships among mean ecological activeness $\theta$, quantities characterizing dynamic ranges of populations $\mathcal{A}$ and $\alpha$, and the ecological force $F_{\alpha}$. The analysis identifies an entire orbit as a stationary ecology, and establishes the notion of an "equation of ecological state". Studies of the stochastic dynamics with finite populations show the LV equation as the rubust, fast cyclic underlying behavior. The mathematical narrative provides a novel way of capturing long-term ecological dynamical behavior with an emergent conservative ecology.
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a thermodynamic theory of ecology Helmholtz Theorem for lotka volterra equation extended conservation law and stochastic predator prey dynamics
arXiv: Mathematical Physics, 2014Co-Authors: Hong QianAbstract:We carry out mathematical analyses, {\em \`{a} la} Helmholtz's and Boltzmann's 1884 studies of monocyclic Newtonian dynamics, for the Lotka-Volterra (LV) equation exhibiting predator-prey oscillations. In doing so a novel "thermodynamic theory" of ecology is introduced. An important feature, absent in the classical mechanics, of ecological systems is a natural stochastic population dynamic formulation of which the deterministic equation (e.g., the LV equation studied) is the infinite population limit. Invariant density for the stochastic dynamics plays a central role in the deterministic LV dynamics. We show how the conservation law along a single trajectory extends to incorporate both variations in a model parameter $\alpha$ and in initial conditions: Helmholtz's Theorem establishes a broadly valid conservation law in a class of ecological dynamics. We analyze the relationships among mean ecological activeness $\theta$, quantities characterizing dynamic ranges of populations $\mathcal{A}$ and $\alpha$, and the ecological force $F_{\alpha}$. The analyses identify an entire orbit as a stationary ecology, and establish the notion of "equation of ecological states". Studies of the stochastic dynamics with finite populations show the LV equation as the robust, fast cyclic underlying behavior. The mathematical narrative provides a novel way of capturing long-term dynamical behaviors with an emergent {\em conservative ecology}.
Yi-an Ma - One of the best experts on this subject based on the ideXlab platform.
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A thermodynamic theory of ecology: Helmholtz Theorem for Lotka–Volterra equation, extended conservation law, and stochastic predator–prey dynamics
Proceedings of The Royal Society A: Mathematical Physical and Engineering Sciences, 2015Co-Authors: Yi-an Ma, Hong QianAbstract:We carry out mathematical analyses, a la Helmholtz’s and Boltzmann’s 1884 studies of monocyclic Newtonian dynamics, for the Lotka–Volterra (LV) equation exhibiting predator–prey oscillations. In doing so, a novel ‘thermodynamic theory’ of ecology is introduced. An important feature, absent in the classical mechanics, of ecological systems is a natural stochastic population dynamic formulation of which the deterministic equation (e.g. the LV equation studied) is the infinite population limit. Invariant density for the stochastic dynamics plays a central role in the deterministic LV dynamics. We show how the conservation law along a single trajectory extends to incorporate both variations in a model parameter α and in initial conditions: Helmholtz’s Theorem establishes a broadly valid conservation law in a class of ecological dynamics. We analyse the relationships among mean ecological activeness θ , quantities characterizing dynamic ranges of populations A and α , and the ecological force F α . The analyses identify an entire orbit as a stationary ecology, and establish the notion of an ‘equation of ecological states’. Studies of the stochastic dynamics with finite populations show the LV equation as the robust, fast cyclic underlying behaviour. The mathematical narrative provides a novel way of capturing long-term dynamical behaviours with an emergent conservative ecology .
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Universal ideal behavior and macroscopic work relation of linear irreversible stochastic thermodynamics
New Journal of Physics, 2015Co-Authors: Yi-an Ma, Hong QianAbstract:We revisit the Ornstein–Uhlenbeck (OU) process as the fundamental mathematical description of linear irreversible phenomena, with fluctuations, near an equilibrium. By identifying the underlying circulating dynamics in a stationary process as the natural generalization of classical conservative mechanics, a bridge between a family of OU processes with equilibrium fluctuations and thermodynamics is established through the celebrated Helmholtz Theorem. The Helmholtz Theorem provides an emergent macroscopic ‘equation of state’ of the entire system, which exhibits a universal ideal thermodynamic behavior. Fluctuating macroscopic quantities are studied from the stochastic thermodynamic point of view and a non-equilibrium work relation is obtained in the macroscopic picture, which may facilitate experimental study and application of the equalities due to Jarzynski, Crooks, and Hatano and Sasa.
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The Helmholtz Theorem for the Lotka-Volterra Equation, the Extended Conservation Relation, and Stochastic Predator-Prey Dynamics
arXiv: Mathematical Physics, 2014Co-Authors: Yi-an Ma, Hong QianAbstract:We carry out a mathematical analysis, \`{a} la Helmholtz's and Boltzmann's 1884 studies of monocyclic Newtonian mechanics, for the Lotka-Volterra (LV) equation exhibiting oscillatory predator-prey dynamics. One of the important features of the latter system, absent in the classical mechanical model, is a natural stochastic dynamic formulation of which the LV equation is the infinite population limit. The invariant density for the stochastic dynamics plays a central role in the deterministic LV dynamics. We show how the conservation law along a single trajectory can be extended to incorporate both variations in model parameter $\alpha$ and in the initial conditions: Helmholtz's Theorem establishes a broadly valid conservation law in a class of ecological dynamics. We analyze the relationships among mean ecological activeness $\theta$, quantities characterizing dynamic ranges of populations $\mathcal{A}$ and $\alpha$, and the ecological force $F_{\alpha}$. The analysis identifies an entire orbit as a stationary ecology, and establishes the notion of an "equation of ecological state". Studies of the stochastic dynamics with finite populations show the LV equation as the rubust, fast cyclic underlying behavior. The mathematical narrative provides a novel way of capturing long-term ecological dynamical behavior with an emergent conservative ecology.
Michele Campisi - One of the best experts on this subject based on the ideXlab platform.
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Microscopic Foundations of Thermodynamics and Generalized Statistical Ensembles
2008Co-Authors: Michele CampisiAbstract:This dissertation aims at addressing two important theoretical questions which are still debated in the statistical mechanical community. The first question has to do with the outstanding problem of how to reconcile time-reversal asymmetric macroscopic laws with the time-reversal symmetric laws of microscopic dynamics. This problem is addressed by developing a novel mechanical approach inspired by the work of Helmholtz on monocyclic systems and the Heat Theorem, i.e., the Helmholtz Theorem. By following a line of investigation initiated by Boltzmann, a Generalized Helmholtz Theorem is stated and proved. This Theorem provides us with a good microscopic analogue of thermodynamic entropy. This is the volume entropy, namely the logarithm of the volume of phase space enclosed by the constant energy hyper-surface. By using quantum mechanics only, it is shown that such entropy can only increase. This can be seen as a novel rigorous proof of the Second Law of Thermodynamics that sheds new light onto the arrow of time problem. The volume entropy behaves in a thermodynamic-like way independent of the number of degrees of freedom of the system, indicating that a whole thermodynamic-like world exists at the microscopic level. It is also shown that breaking of ergodicity leads to microcanonical phase transitions associated with nonanalyticities of volume entropy. The second part of the dissertation deals with the problem of the foundations of generalized ensembles in statistical mechanics. The starting point is Boltzmann's work on statistical ensembles and its relation with the Heat Theorem. We first focus on the nonextensive thermostatistics of Tsallis and the associated deformed exponential ensembles. These ensembles are analyzed in detail and proved (a) to comply with the requirements posed by the Heat Theorem, and (b) to interpolate between canonical and microcanonical ensembles. Further they are showed to describe finite systems in contact with finite heat baths. Their mechanical and information-theoretic foundation, are highlighted. Finally, a wide class of generalized ensembles is introduced, all of which reproduce the Heat Theorem. This class, named the class of dual orthodes, contains microcanonical, canonical, Tsallis and Gaussian ensembles as special cases.
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statistical mechanical proof of the second law of thermodynamics based on volume entropy
Studies in History and Philosophy of Modern Physics, 2008Co-Authors: Michele CampisiAbstract:Abstract In a previous work [Campisi, M. (2005). On the mechanical foundations of thermodynamics: The generalized Helmholtz Theorem. Studies in History and Philosophy of Modern Physics, 36, 275–290] we have addressed the mechanical foundations of equilibrium thermodynamics on the basis of the generalized Helmholtz Theorem. It was found that the volume entropy provides a good mechanical analogue of thermodynamic entropy because it satisfies the heat Theorem and it is an adiabatic invariant. This property explains the “equal” sign in Clausius principle ( S f ⩾ S i ) in a purely mechanical way and suggests that the volume entropy might explain the “larger than” sign (i.e. the law of entropy increase) if non-adiabatic transformations are considered. Based on the principles of microscopic (quantum or classical) mechanics we prove here that, provided the initial equilibrium satisfies the natural condition of decreasing ordering of probabilities, the expectation value of the volume entropy cannot decrease for arbitrary transformations performed by some external source of work on an insulated system. This can be regarded as a rigorous quantum-mechanical proof of the second law. We discuss how this result relates to the minimal work principle and how it improves on previous attempts. The natural evolution of entropy is towards larger values because the natural state of matter is at positive temperature. Actually the law of entropy decrease holds in artificially prepared negative temperature systems.
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statistical mechanical proof of the second law of thermodynamics based on volume entropy
Studies in History and Philosophy of Modern Physics, 2008Co-Authors: Michele CampisiAbstract:In a previous work (M. Campisi. Stud. Hist. Phil. M. P. 36 (2005) 275-290) we have addressed the mechanical foundations of equilibrium thermodynamics on the basis of the Generalized Helmholtz Theorem. It was found that the volume entropy provides a good mechanical analogue of thermodynamic entropy because it satisfies the heat Theorem and it is an adiabatic invariant. This property explains the ``equal'' sign in Clausius principle ($S_f \geq S_i$) in a purely mechanical way and suggests that the volume entropy might explain the ``larger than'' sign (i.e. the Law of Entropy Increase) if non adiabatic transformations were considered. Based on the principles of microscopic (quantum or classical) mechanics here we prove that, provided the initial equilibrium satisfy the natural condition of decreasing ordering of probabilities, the expectation value of the volume entropy cannot decrease for arbitrary transformations performed by some external sources of work on a insulated system. This can be regarded as a rigorous quantum mechanical proof of the Second Law. We discuss how this result relates to the Minimal Work Principle and improves over previous attempts. The natural evolution of entropy is towards larger values because the natural state of matter is at positive temperature. Actually the Law of Entropy Decrease holds in artificially prepared negative temperature systems.
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On the mechanical foundations of thermodynamics: The generalized Helmholtz Theorem
Studies in History and Philosophy of Modern Physics, 2005Co-Authors: Michele CampisiAbstract:An elegant but seldom appreciated effort to provide a mechanical model of equilibrium thermodynamics dates back to the Helmholtz Theorem (HT). According to this Theorem, the thermodynamic relations hold mechanically (without probabilistic assumptions) in the case of one-dimensional monocyclic systems. Thanks to a discrete picture of the phase space, Boltzmann was able to apply the HT to multi-dimensional ergodic systems, suggesting that the thermodynamic relations we observe in macroscopic systems at equilibrium are a direct consequence of the microscopic laws of dynamics alone. Here I review Boltzmann’s argument and show that, using the language of the modern ergodic theory, it can be safely re-expressed on a continuumphase space as a generalized Helmholtz Theorem(GHT), which can be readily proved. Along the way the agreement between the Helmholtz–Boltzmann theory and that of P. Hertz (based on adiabatic invariance) is revealed. Both theories, in fact, lead to define the entropy as the logarithmof the phase-space volum e enclosed by the constant energy hypersurface (volume entropy). r 2005 Elsevier Ltd. All rights reserved.
Andrew Stewart - One of the best experts on this subject based on the ideXlab platform.
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Does the Helmholtz Theorem of vector decomposition apply to the wave fields of electromagnetic radiation
Physica Scripta, 2014Co-Authors: Andrew StewartAbstract:The derivation of the Helmholtz Theorem of vector decomposition of a 3-vector field requires that the field satisfy certain convergence properties at spatial infinity. This paper investigates if time-dependent electromagnetic radiation wave fields, which are of long range, satisfy these requirements. It is found that the requirements are satisfied because the fields give rise to integrals over the radial distance r of integrands of the form sin(kr)/r and cos(kr)/r. These Dirichlet integrals converge at infinity as required.
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Does the Helmholtz Theorem of vector decomposition apply to the wave fields of electromagnetic radiation
Physica Scripta, 2014Co-Authors: Andrew StewartAbstract:The derivation of the Helmholtz Theorem of vector decomposition of a three-vector field requires that the field satisfy certain convergence properties at spatial infinity. This paper investigates if time-dependent electromagnetic radiation wave fields of point sources, which are of long range, satisfy these requirements. It is found that the requirements are satisfied because the fields give rise to integrals over the radial distance r of integrands of the form sin(kr)/r and cos(kr)/r. These Dirichlet integrals converge at infinity as required.
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Longitudinal and transverse components of a vector field
Sri Lankan Journal of Physics, 2012Co-Authors: Andrew StewartAbstract:A unified account, from a pedagogical perspective, is given of the longitudinal and transverse projective delta functions proposed by Belinfante and of their relation to the Helmholtz Theorem for the decomposition of a vector field into its longitudinal and transverse components. It is argued that the results are applicable to fields that are time-dependent as well as fields that are time-independent. Keywords: Classical electromagnetism; Helmholtz's Theorem; transverse; longitudinal DOI: http://dx.doi.org/10.4038/sljp.v12i0.3504 Sri Lankan Journal of Physics, Vol. 12 (2011) 33-42
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Orbital and spin components of the angular momentum of a general electromagnetic field
arXiv: Classical Physics, 2006Co-Authors: Andrew StewartAbstract:A decomposition of the angular momentum of the classical electromagnetic field into orbital and spin components that is manifestly gauge invariant and general has been obtained. This is done by decomposing the electric field into its longitudinal and transverse parts by means of the Helmholtz Theorem. The orbital and spin components of the angular momentum of any specified electromagnetic field can be found from this prescription.
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Angular Momentum of Light
Journal of Modern Optics, 2005Co-Authors: Andrew StewartAbstract:By means of the Helmholtz Theorem on the decomposition of vector fields, the angular momentum of the classical electromagnetic field is decomposed, in a general and manifestly gauge invariant manner, into a spin component and an orbital component. The method is applied to linearly and circularly polarized plane waves in their classical and quantum forms.
Andrew Chubykalo - One of the best experts on this subject based on the ideXlab platform.
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Helmholtz Theorems, Gauge Transformations, General Covariance and the Empirical Meaning of Gauge Conditions
Journal of Modern Physics, 2016Co-Authors: Andrew Chubykalo, Augusto Espinoza, Rolando Alvarado FloresAbstract:It is well known that the use of Helmholtz decomposition Theorem for static vector fields , when applied to the time dependent vector fields , which represent the electromagnetic field, allows us to obtain instantaneous-like solutions all along . For this reason, some people thought (see e.g. [1] and references therein) that the Helmholtz Theorem cannot be applied to time dependent vector fields and some modification is wanted in order to get the retarded solutions. However, the use of the Helmholtz Theorem for static vector fields is correct even for time dependent vector fields (see, e.g. [2]), so a relation between the solutions was required, in such a way that a retarded solution can be transformed in an instantaneous one, and conversely. On this paper we want to suggest, following most of the time the mathematical formalism of Woodside in [3], that: 1) there are many Helmholtz decompositions, all equally consistent, 2) each one is naturally related to a space-time structure, 3) when we use the Helmholtz decomposition for the electromagnetic potentials it is equivalent to a gauge transformation, 4) there is a natural methodological criterion for choosing the gauge according to the structure postulated for a global space-time, 5) the Helmholtz decomposition is the manifestation at the level of the fields that a gauge is involved. So, when we relate the retarded solution to the instantaneous one what we do is to change the gauge and the space-time. And, if the Helmholtz decompositions are related to a space-time structure, and are equivalent to gauge transformations, each gauge transformation is natural for a specific space-time. In this way, a Helmholtz decomposition for Euclidean space is equivalent to the Coulomb gauge and a Helmholtz decomposition for the Minkowski space is equivalent to the Lorenz gauge. This leads us to consider that the theories defined by different gauges may be mathematically equivalent, because they can be related by means of a gauge transformation, but they are not empirically equivalent, because they have quite different observational consequences due to the different space-time structure involved.
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Relativistic analysis of application of Helmholtz Theorem to classical electrodynamics
Journal of Advances in Physics, 2015Co-Authors: Andrew Chubykalo, Rolando Alvarado-flores, Augusto EspinozaAbstract:In this work we discuss the relationship between the instantaneous-action-at-a-distance solutions of Maxwell’s equations obtained using Helmholtz Theorem and the Lorentz’s invariant solutions of the same equations obtained using Special Relativity postulates. We show that Special Relativity postulates are not consistent with Helmholtz’s Theorem in the presence of charges and currents, but in the vacuum, without charges and currents, Helmholtz’s Theorem and Special Relativity agree because the instantaneous-action-at-a-distance solutions can be eliminated using a gauge transformation.
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Reply to ‘Comment on “Helmholtz Theorem and the V-Gauge in the Problem of Superluminal and Instantaneous Signals in Classical Electrodynamics” by A. Chubykalo et al.’ by J.A. Heras [Found. Phys. Lett. vol. 19(6) p. 579 (2006)]
Foundations of Physics, 2007Co-Authors: Andrew Chubykalo, Augusto Espinoza, R. Alvarado Flores, A. Gutierrez RodriguezAbstract:This is the reply to `COMMENT ON ``Helmholtz Theorem AND THE V-GAUGE IN THE PROBLEM OF SUPERLUMINAL AND INSTANTANEOUS SIGNALS IN CLASSICAL ELECTRODYNAMICS" BY A. CHUBYKALO ET AL' BY J. A. HERAS [FOUND. PHYS. LETT. vol. 19(6) p. 579 (2006)]
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reply to comment on Helmholtz Theorem and the v gauge in the problem of superluminal and instantaneous signals in classical electrodynamics by a chubykalo et al by j a heras found phys lett vol 19 6 p 579 2006
arXiv: General Physics, 2007Co-Authors: Andrew Chubykalo, Augusto Espinoza, R Alvaradoflores, Gutierrez A RodriguezAbstract:This is the reply to `COMMENT ON ``Helmholtz Theorem AND THE V-GAUGE IN THE PROBLEM OF SUPERLUMINAL AND INSTANTANEOUS SIGNALS IN CLASSICAL ELECTRODYNAMICS" BY A. CHUBYKALO ET AL' BY J. A. HERAS [FOUND. PHYS. LETT. vol. 19(6) p. 579 (2006)]
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Helmholtz Theorem and the V-Gauge in the Problem of Superluminal and Instantaneous Signals in Classical Electrodynamics
Foundations of Physics Letters, 2006Co-Authors: Andrew Chubykalo, Augusto Espinoza, R. Alvarado Flores, A. Gutierrez RodriguezAbstract:In this work we substantiate the applying of the Helmholtz vector decomposition Theorem (H-Theorem) to vector fields in classical electrodynamics. Using the H-Theorem, within the framework of the two-parameter Lorentz-like gauge (so called v-gauge ), we show that two kinds of magnetic vector potentials exist: one of them (solenoidal) can act exclusively with the velocity of light c and the other one (irrotational) with an arbitrary finite velocity v (including a velocity more than c ). We show also that the irrotational component of the electric field has a physical meaning and can propagate exclusively instantaneously .