Local Energy Conservation Research Articles

Comparing matching prescriptions between pre-equilibrium and hydrodynamic models in high-energy nuclear collisions

State-of-the-art simulations of high-energy nuclear collisions rely on hybrid setups, involving in particular a pre-equilibrium stage to let the system evolve from a far-from-equilibrium initial condition towards a near-equilibrated state after which fluid dynamics can be applied meaningfully. A known issue is the mismatch between the equation of state in the fluid-dynamical evolution and the effective one in the previous stage, which leads to discontinuities at the interface between the two models. Here we introduce a new matching prescription at this interface, based on the entropy, and we compare it with the standard one relying on local energy conservation. We study the behavior of various quantities at the switching time between the models and investigate a number of final-state hadronic observables. For the latter, we show that they are not modified significantly by the choice of matching prescription, provided an appropriate normalization is chosen for the initial state. In turn, our approach reduces sizeably the ratio of bulk over thermodynamic pressure at the beginning of the fluid-dynamical stage.

A Local Macroscopic Conservative (LoMaC) Low Rank Tensor Method for the Vlasov Dynamics

In this paper, we propose a novel Local Macroscopic Conservative (LoMaC) low rank tensor method for simulating the Vlasov-Poisson (VP) system. The LoMaC property refers to the exact local conservation of macroscopic mass, momentum and energy at the discrete level. This is a follow-up work of our previous development of a conservative low rank tensor approach for Vlasov dynamics (arXiv:2201.10397). In that work, we applied a low rank tensor method with a conservative singular value decomposition to the high dimensional VP system to mitigate the curse of dimensionality, while maintaining the local conservation of mass and momentum. However, energy conservation is not guaranteed, which is a critical property to avoid unphysical plasma self-heating or cooling. The new ingredient in the LoMaC low rank tensor algorithm is that we simultaneously evolve the macroscopic conservation laws of mass, momentum and energy using a flux-difference form with kinetic flux vector splitting; then the LoMaC property is realized by projecting the low rank kinetic solution onto a subspace that shares the same macroscopic observables by a conservative orthogonal projection. The algorithm is extended to the high dimensional problems by hierarchical Tuck decomposition of solution tensors and a corresponding conservative projection algorithm. Extensive numerical tests on the VP system are showcased for the algorithm’s efficacy.

High order energy-preserving method for the space fractional Klein–Gordon-Zakharov equations

The space fractional Klein–Gordon-Zakharov equations are transformed into the multi-symplectic structure system by introducing new auxiliary variables. The multi-symplectic system, which satisfies the multi-symplectic conservation, local energy and momentum conservation, is discretizated into the semi-discrete multi-symplectic system by the Fourier pseudo-spectral method. The second order multi-symplectic average vector field method is applied to the semi-discrete system. The fully discrete energy preserving scheme of the space fractional Klein–Gordon-Zakharov equation is obtained. Based on the composition method, a fourth order energy preserving scheme of the Riesz space fractional Klein–Gordon-Zakharov equations is also obtained. Numerical experiments confirm that these new schemes can have computing ability for a long time and can well preserve the discrete energy conservation property of the equations.

Toward local Madelung mechanics in spacetime

It has recently been shown that relativistic quantum theory leads to a local interpretation of quantum mechanics wherein the universal wavefunction in configuration space is entirely replaced with an ensemble of local fluid equations in spacetime. For want of a fully relativistic quantum fluid treatment, we develop a model using the nonrelativistic Madelung equations, and obtain conditions for them to be local in spacetime. Every particle in the Madelung fluid is equally real, and has a definite position, momentum, kinetic energy, and potential energy. These are obtained by defining quantum momentum and kinetic energy densities for the fluid and separating the momentum into average and symmetric parts, and kinetic energy into classical kinetic and quantum potential parts. The two types of momentum naturally give rise to a single classical kinetic energy density, which contains the expected kinetic energy, even for stationary states, and we define the reduced quantum potential as the remaining part of the quantum kinetic energy density. We treat the quantum potential as a novel mode of internal energy storage within the fluid particles, which explains most of the nonclassical behavior of the Madelung fluid. For example, we show that in tunneling phenomena, the quantum potential negates the barrier so that nothing prevents the fluid from flowing through. We show how energy flows and transforms in this model, and that enabling local conservation of energy requires defining a quantum potential energy current that flows through the fluid rather than only flowing with it. The nonrelativistic treatment generally contains singularities in the velocity field, which undermines the goal of local dynamics, but we expect a proper relativistic treatment will bound the fluid particle velocities at c.

Is the Local Energy Conservation Law Valid inside the Electronic Device of Charged Capacitors?

The energy conservation is one of the most fundamental and well established principles in physics. Emmy Noether extended the energy conservation principle to the quantum field theoretical domain in empty space by relating the time translation invariance of the universe with energy conservation. While this is the case in open empty space, it seems that the local space enclosed by conducting metallic plates has unexpected property suggesting that the energy conservation principle does not necessarily apply to localized bound system of capacitors in electrodynamics. This point of view was raised by noticing the fact that the spherical capacitor has calculable electrostatic self-potential energy in both the inner and outer shells respectively which was not taken into account in the conventional consideration of the total energy stored in the capacitors. It seems like the concept of moving charges one by one into the capacitor plates has blindsided the investigators to bypass the necessary steps to account for the additional repulsive self-potential energy that accumulates simultaneously in both of the capacitor plates in the process of charging the capacitor. The investigation shows that the repulsive self-potential energies from both of the capacitor plates have indeed been omitted in the conventional calculation of the stored energy in the capacitors. We present the dynamics of the stored potential energy in capacitors and discuss its physical consequences in relation to the anomalous energy devices reported in the past.

Edge of chaos as critical local symmetry breaking in dissipative nonautonomous systems

The fully nonlinear notion of resonance−geometrical resonance−in the general context of dissipative systems subjected to spatially periodic phase-modulated potentials is discussed. It is demonstrated that there is an exact local invariant associated with each geometrical resonance solution which reduces to the system’s energy when the potential is stationary. The geometrical resonance solutions represent a local symmetry whose critical breaking leads to a new analytical criterion for the onset of chaotic instabilities. This physical criterion is deduced in the co-moving frame from the local energy conservation over the shortest significant timescale. Remarkably, the new physical criterion for the onset of chaotic instabilities is shown to be valid over large regions of parameter space, thus being useful beyond the scope of current mathematical techniques. More importantly, the present theory helps to understand the unreasonable effectiveness of the Melnikov’s method beyond the perturbative regime.

Exponentially fitted methods with a local energy conservation law

A new exponentially fitted version of the discrete variational derivative method for the efficient solution of oscillatory complex Hamiltonian partial differential equations is proposed. When applied to the nonlinear Schrödinger equation, this scheme has discrete conservation laws of charge and energy. The new method is compared with other conservative schemes from the literature on a benchmark problem whose solution is an oscillatory breather wave.

Gravitation with modified fluid Lagrangian: Variational principle and an early dark energy model

Variational principle is the main approach to obtain complete and self-consistent field equations in gravitational theories. This method works well in pure field cases such as $f(R)$ and Horndeski gravities. However, debates exist in the literature over the modification of perfect fluid. This paper aims to clarify this issue. For a wide class of modified fluid Lagrangian, we show that the variational principle is unable to give complete field equations. One additional equation is required for completeness. Adopting the local energy conservation equation gives the modified fluid a good thermodynamic interpretation. Our result is the first modified fluid theory that can incorporate energy conservation. As an application of this framework, we propose a specific modified fluid model to realize early dark energy triggered by cosmic radiation-matter transition. This model naturally explains why early dark energy occurs around matter-radiation equality and is useful in erasing the Hubble tension.

Friedmann equations and cosmic bounce in a modified cosmological scenario

In this work we present a derivation of modified Raychaudhuri and Friedmann equations from a phenomenological model of quantum gravity based on the thermodynamics of spacetime. Starting from general gravitational equations of motion which encode low-energy quantum gravity effects, we found its particular solution for homogeneous and isotropic universes with standard matter content, obtaining a modified Raychaudhuri equation. Then, we imposed local energy conservation and used a perturbative treatment to derive a modified Friedmann equation. The modified evolution in the early universe we obtained suggests a replacement of the Big Bang singularity by a regular bounce. Lastly, we also briefly discuss the range of validity of the perturbative approach and its results.

Constraining Co-Varying Coupling Constants from Globular Cluster Age

Equations governing the evolution of a star involve multiple coupling constants. Thus, the time it spends as a main-sequence star can be expected to depend on whether or not such constants vary over the time scale of stellar evolution. When the star belongs to a globular cluster, the star’s age cannot exceed that of the globular cluster, and the latter cannot exceed the age of the Universe. This fact can be used to constrain or verify the variation of the coupling constants, i.e., the speed of light c, the gravitational constant G, the Planck constant h, and the Boltzmann constant k. We have estimated the age of the main-sequence star analytically from the time it takes to synthesize all its hydrogen into helium under fixed and varying coupling constants scenarios. When we permitted the interrelated variation of the four constants (G~c3~h3~k3/2) and differentiated between the cosmological energy and local energy conservation laws, we could show that the variation of the constants established in our earlier studies, i.e., G˙/G=3c˙/c=3h˙/h=1.5k˙/k=3.90±0.04×10−10 yr−1 at the current cosmic time is consistent with the present work. Nevertheless, the challenge remains to come up with an experiment, astrometric or terrestrial, that can unequivocally prove or falsify the predicted variation.

Global and local conservation of mass, momentum and kinetic energy in the simulation of compressible flow

The spatial discretization of convective terms in compressible flow equations is studied from an abstract viewpoint, for finite-difference methods and finite-volume type formulations with cell-centered numerical fluxes. General conditions are sought for the local and global conservation of primary (mass and momentum) and secondary (kinetic energy) invariants on Cartesian meshes. The analysis, based on a matrix approach, shows that sharp criteria for global and local conservation can be obtained and that in many cases these two concepts are equivalent. Explicit numerical fluxes are derived in all finite-difference formulations for which global conservation is guaranteed, even for non-uniform Cartesian meshes. The treatment reveals also an intimate relation between conservative finite-difference formulations and cell-centered finite-volume type approaches. This analogy suggests the design of wider classes of finite-difference discretizations locally preserving primary and secondary invariants.

The Impact of Extremely Low Electric Rates on Energy Conservation, Planning and Resilience: A Case Study of Plattsburgh, New York

Electric rates in Plattsburgh, New York are among the least expensive in the United States. Plattsburgh’s rates are set by a preference power program that was enacted in 1957 to promote rural economic development. A survey distributed to customers of Plattsburgh’s municipally-owned electric utility inquired about familiarity with local electric rates and energy systems, conservation behaviors, and attitudes towards policy interventions intended to encourage conservation. Results indicate that energy consumers in Plattsburgh hold a moderate degree of familiarity with the community’s energy system and implement strong conservation actions. Support for policy interventions are mixed. Survey results reveal income inequality, as differences in conservation attitudes and behaviors are apparent among respondents that report varying degrees of financial vulnerability. Today, Plattsburgh is in need of a resilient energy system, yet inexpensive rates discourage investment in infrastructure. This research reveals the complex social, economic and policy forces interacting to determine Plattsburgh’s energy future.

Dynamical Symmetry Breaking of Infinite-Dimensional Stochastic System

The mapping relationship between the symmetry and the conserved quantity inspired researchers to seek the conserved quantity from the viewpoint of the symmetry for the dynamic systems. However, the symmetry breaking in the dynamic systems is more common than the symmetry in the engineering. Thus, as the supplement of our previous work on the symmetry breaking of infinite-dimensional deterministic dynamic systems, the dynamical symmetry breaking of infinite-dimensional stochastic systems is discussed in this paper. Following a brief review of the stochastic (multi-)symplectic for the dynamic system excited by stochastic white noise, two types of stochastic symmetry breaking factors, including the general stochastic excitation and the general stochastic parameters of the infinite-dimensional dynamic systems, are investigated in detail. We find that both the general stochastic excitation and the general stochastic parameters will not break the local multi-symplectic structure of the dynamic systems. However, the local energy conservation law will be broken by the general stochastic excitation, as well as by the stochastic parameters, which are given by the local energy dissipation in this paper. To illustrate the validity of the analytical results, the stochastic vibration of a clamped single-walled carbon nanotube is investigated and the critical condition of the appearance of chaos is obtained. The theoretical results obtained can be used to guide us to construct the structure-preserving method for the stochastic dynamic systems.

How energy is conserved in Newtonian gravity

The concept underlying conservation of energy in a force field is that the work done by particles moving against the force represents energy removed from particles and added to energy in the field. For electromagnetic fields and charged particles, the transfer of energy between charges and local fields is described by Poynting's theorem. However, a similar local model for gravitational energy is not usually taught. In this paper, I derive the equations describing local conservation of energy in Newtonian gravity, discuss the well-known non-uniqueness of various terms, and show how comparison with Poynting's theorem provides insight into conditions for physical significance.

The exponential invariant energy quadratization approach for general multi-symplectic Hamiltonian PDEs

Many conservative systems in physical sciences can be described by multi-symplectic Hamiltonian PDEs (MS-HPDEs) admitting three important inherent properties named as multi-symplectic conservation law, local energy conservation law and local momentum conservation law. In this paper, we develop a novel strategy to systematically derive linearly implicit local energy-preserving schemes for general MS-HPDEs, which is named the exponential invariant energy quadratization approach (EIEQ). Such novel strategy is based on the exponential form of non-quadratic terms of the state function that can remove the bounded-from-below restriction, so it is more applicable than the traditional IEQ approach widely employed to construct linearly implicit schemes. Moreover, we provide a completely explicit discretization of the auxiliary variable combined with the nonlinear term, which obtains linearly implicit schemes of the constant coefficient, making their more effective than the IEQ schemes for rapid simulations. We also present the multiple EIEQ approach to improve the applicability of the EIEQ approach for MS-HPDEs with more non-quadratic terms. In addition, when periodic or homogeneous boundary conditions are considered, the proposed schemes are global energy-preserving and can be explicitly solved by using the fast Fourier transform. Finally, ample numerical tests are carried out to demonstrate the computational efficiency, conservation and accuracy of EIEQ schemes.

L∞ bounds for Chern–Simons gauged equations in R1+1 and their applications

We estimate the L∞ norm of the first derivative of the solution to the Chern–Simons gauged O(3) sigma model in R1+1 by using local energy conservation and observing the transport equation along the characteristic line. As an application of the bound, we improve the growth rate of the H2 norm of the solution from an exponential to a polynomial one. We also apply a similar idea to the Chern–Simons–Higgs model.

Explicit structure-preserving geometric particle-in-cell algorithm in curvilinear orthogonal coordinate systems and its applications to whole-device 6D kinetic simulations of tokamak physics

Explicit structure-preserving geometric particle-in-cell (PIC) algorithm in curvilinear orthogonal coordinate systems is developed. The work reported represents a further development of the structure-preserving geometric PIC algorithm achieving the goal of practical applications in magnetic fusion research. The algorithm is constructed by discretizing the field theory for the system of charged particles and electromagnetic field using Whitney forms, discrete exterior calculus, and explicit non-canonical symplectic integration. In addition to the truncated infinitely dimensional symplectic structure, the algorithm preserves exactly many important physical symmetries and conservation laws, such as local energy conservation, gauge symmetry and the corresponding local charge conservation. As a result, the algorithm possesses the long-term accuracy and fidelity required for first-principles-based simulations of the multiscale tokamak physics. The algorithm has been implemented in the SymPIC code, which is designed for high-efficiency massively-parallel PIC simulations in modern clusters. The code has been applied to carry out whole-device 6D kinetic simulation studies of tokamak physics. A self-consistent kinetic steady state for fusion plasma in the tokamak geometry is numerically found with a predominately diagonal and anisotropic pressure tensor. The state also admits a steady-state sub-sonic ion flow in the range of 10 km s−1, agreeing with experimental observations and analytical calculations Kinetic ballooning instability in the self-consistent kinetic steady state is simulated. It is shown that high-n ballooning modes have larger growth rates than low-n global modes, and in the nonlinear phase the modes saturate approximately in 5 ion transit times at the 2% level by the E × B flow generated by the instability. These results are consistent with early and recent electromagnetic gyrokinetic simulations.

A linearized energy-conservative scheme for two-dimensional nonlinear Schrödinger equation with wave operator

Based on the invariant energy quadratization approach, we propose a linear implicit and local energy preserving scheme for the nonlinear Schrödinger equation with wave operator, that describes the solitary waves in physics. In order to overcome the difficulty of designing an efficient scheme for the imaginary functions of the nonlinear Schrödinger equation with wave operator, we transform the original problem into its real form. By introducing some auxiliary variables, the real form of nonlinear Schrödinger equation with wave operator is reformulated into an equivalent system, which admits the modified local energy conservation law. Then the equivalent system is discretized by the finite difference method to yield a linear system at each time step, which can be efficiently solved. A numerical analysis of the proposed scheme is conducted to show its uniquely solvability and convergence. Our proposed method is validated by numerical simulations in terms of accuracy, energy conservation law and stability.

Droplet freezing phase transition and heat transfer under the ultrasonic effect

Ultrasound-assisted freezing technology is getting more attention among researchers due to its outstanding behavior in improving freezing performance. However, the lack of reliable thermo-physical model has prevented its widespread usage in related fields. As demonstrated in this study, a physical model of frozen droplet was established based on the local conservation of energy, which presented the changing rules of droplet's thermal-mass properties including porosity, exchanged heat, and liquid proportion. The variation of the droplet's liquid proportion with no ultrasound, time-controlled ultrasound, and whole-process ultrasound was analyzed, respectively. According to the calculation results, the loading of ultrasound in the droplet's frozen process had the potential to considerably increase the phase transition rate. In particular, the droplets' completely freezing time with the loading of time-controlled ultrasound, whole-process ultrasound, and without ultrasound was 13.5 s, 14.1 s, and 14.6 s, respectively. The maximum relative error of mathematical and experimental results was 6.16%, and the minimum error was 2.96%. This paper contributes to a comprehensive understanding of the ultrasonic effect in improving freezing efficiency and controlling the phase transition process.

Energy conservation for the nonhomogeneous incompressible ideal Hall-MHD equations

In this paper, we study the energy conservation for the nonhomogeneous incompressible ideal Hall-magnetohydrodynamic system. Three types of sufficient conditions are obtained. Precisely, the first one provides ρ, u, P, and b with sufficient regularity to ensure the local energy conservation. The second one removes the regularity condition on P while requires Lp regularity on the spatial gradient of the density ∇ρ and Lr regularity on ρt. The last one removes the regularity condition on ρt while requires certain time regularity on the velocity field u. Our main strategy relies on commutator estimates in the work of Constantin et al. [Commun. Math. Phys. 165, 207–209 (1994)].