On the phenomenological thermodynamics of moving matter

Dantzig, D. van

doi:10.1016/s0031-8914(39)90072-8

Cited by 44 publications

(26 citation statements)

References 11 publications

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“…Nonrelativistic generalizations of the standard theory as, for example, anomalous diffusion processes have been summarized in [83,116,117], while review articles on nonrelativistic quantum Brownian motion can be found in [107,108,118,119]. 5 See also Pauli [123], Eddington [124], Tolman [125] and van Dantzig [126] for early discussions of this problem.…”

Section: Historical Backgroundmentioning

confidence: 99%

Relativistic Brownian motion

2009

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a b s t r a c tOver the past one hundred years, Brownian motion theory has contributed substantially to our understanding of various microscopic phenomena. Originally proposed as a phenomenological paradigm for atomistic matter interactions, the theory has since evolved into a broad and vivid research area, with an ever increasing number of applications in biology, chemistry, finance, and physics. The mathematical description of stochastic processes has led to new approaches in other fields, culminating in the path integral formulation of modern quantum theory. Stimulated by experimental progress in high energy physics and astrophysics, the unification of relativistic and stochastic concepts has re-attracted considerable interest during the past decade. Focusing on the framework of special relativity, we review, here, recent progress in the phenomenological description of relativistic diffusion processes. After a brief historical overview, we will summarize basic concepts from the Langevin theory of nonrelativistic Brownian motions and discuss relevant aspects of relativistic equilibrium thermostatistics. The introductory parts are followed by a detailed discussion of relativistic Langevin equations in phase space. We address the choice of time parameters, discretization rules, relativistic fluctuationdissipation theorems, and Lorentz transformations of stochastic differential equations. The general theory is illustrated through analytical and numerical results for the diffusion of free relativistic Brownian particles. Subsequently, we discuss how Langevin-type equations can be obtained as approximations to microscopic models. The final part of the article is dedicated to relativistic diffusion processes in Minkowski spacetime. Since the velocities of relativistic particles are bounded by the speed of light, nontrivial relativistic Markov processes in spacetime do not exist; i.e., relativistic generalizations of the nonrelativistic diffusion equation and its Gaussian solutions must necessarily be non-Markovian. We compare different proposals that were made in the literature and discuss their respective benefits and drawbacks. The review concludes with a summary of open questions, which may serve as a starting point for future investigations and extensions of the theory.

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Section: Historical Backgroundmentioning

confidence: 99%

Relativistic Brownian motion

2009

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“…in terms of the mass density , the velocity u, the specific entropy s, the three Clebsch potentials [7] ζ, α, β and an additional potential ϑ, termed as thermasy by by Van Dantzig [8]. Furthermore, e( , s) denotes the specific inner energy, η the shear viscosity, η the volume viscosity of the fluid and D :=…”

Section: Conventional Approachmentioning

confidence: 99%

A non‐conventional discontinuous Lagrangian for viscous flow

Scholle

Marner

2016

Proc Appl Math and Mech

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Although many attempts for finding a variational formulation of Navier-Stokes equations have been made, a Lagrangian for viscous flow has not been established yet. An auspicious suggestion was made by Scholle [1] by extending Seliger and Whitham's Lagrangian [2] with additional terms ending up with some partial success: on the one hand, the phenomenon 'viscosity' occurs in a qualitatively correct manner, on the other hand the equations of motion resulting from the variation of Hamilton's principle differ from Navier-Stokes equations and therefore, their solutions reveal noticeable quantitative differences to those of Navier-Stokes equations. In this paper the Lagrangian [1] is modified by applying an innovative idea by Anthony [3], motivated by the reformulation of the Lagrangian in terms of complex fields, which can also be understood as the inversion of Madelungs idea [4] of reformulating the complex Schrödinger's equation into a hydrodynamic form. The prize one has to pay is that the resulting Lagrangian is discontinuous and therefore the mathematical treatment of the related variational problem challenging. Furthermore, an additional parameter, ω0, has to be introduced. However, it is demonstrated that Navier-Stokes equations are recovered by the limit ω0 → ∞, whereas the case of finite ω0 can be interpreted as a generalization towards non-equilibrium thermodynamics [3]. in terms of the mass density , the velocity u, the specific entropy s, the three Clebsch potentials with kinematic viscosity ν := η/ 0 . These differ significantly from Navier-Stokes equations: on the one hand they are third order PDE, on the other hand the thermasy ϑ appears explicitly in the field equations as an additional degree of freedom. a unidirectional flow geometry u = u(y) e x and considering no-slip conditions u(0) = u(h) = 0 at lower and upper plate, the classical parabolic flow profile, u(y) = Ky(h − y)/2, is recovered if the solution ϑ/T = t of Eq. (4) is considered for

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“…[11]) for a nonisentropic perfect fluid with equation of state ρ(n, J) = nm(J) + w(n). In the perfect fluid literature, the variable J is interpreted as the entropy per particle and the variable Θ is interpreted as the thermasy [12]. (The thermasy is a variable whose gradient along the particle worldlines is proportional to the local temperature.)…”

Section: A Coupling Clocksmentioning

confidence: 99%

“…We show the inequivalence of this clock coupling to the coupling used by DeWitt [3], and discuss the advantages of DeWitt's method. We also point out that for a nonisentropic perfect fluid S, the thermasy [12] can be reinterpreted as a clock variable with DeWitt-type coupling. Then, in preparation for Sec.…”

Section: Introductionmentioning

confidence: 99%

Relativistic material reference systems

Brown

Marolf²

1996

Phys. Rev. D

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This work closes certain gaps in the literature on material reference systems in general relativity. It is shown that perfect fluids are a special case of DeWitt's relativistic elastic media and that the velocity-potential formalism for perfect fluids can be interpreted as describing a perfect fluid coupled to a fleet of clocks. A Hamiltonian analysis of the elastic media with clocks is carried out and the constraints that arise when the system is coupled to gravity are studied. When the Hamiltonian constraint is resolved with respect to the clock momentum, the resulting true Hamiltonian is found to be a functional only of the gravitational variables. The true Hamiltonian is explicitly displayed when the medium is dust, and is shown to depend on the detailed construction of the clocks.

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On the phenomenological thermodynamics of moving matter

Cited by 44 publications

References 11 publications

Relativistic Brownian motion

Relativistic Brownian motion

A non‐conventional discontinuous Lagrangian for viscous flow

Relativistic material reference systems

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