Potential game is an emerging notion and framework for studying N-player games, especially with heterogeneous players. In this paper, we build an analytical framework for dynamic potential games. We prove that a game is a dynamic potential game if and only if each player's value function can be decomposed as a potential function and a residual term which is solely dependent on other players' policies. This decomposition is consistent with the result in the static setting and enables us to identify and analyze an important and new class of dynamic potential games called the distributed game. Moreover, we prove that a game is a dynamic potential game if the value function has a symmetric Jacobian. This generalizes the differential characterization for static potential games by replacing the classical derivative with a new notation of functional derivative with respect to Markov policies. For a general class of continuous-time stochastic games, we explicitly characterize their potential functions. In particular, we show that the potential function of linear-quadratic games can be studied through a system of linear ODEs. Furthermore, under a rank condition on control coefficients, we p
We propose a new, general form for a pseudo-Newtonian gravitational potential (PNP), expressed as a series of Paczyński-Wiita-like functions with the addition of increasing negative powers of $r$ with arbitrary coefficients. We present a procedure for determining these coefficients to construct a custom PNP that replicates key features of Schwarzschild geodesics for a test particle near a black hole. As an example, we construct potentials set to reproduce (I) the presence of an innermost stable circular orbit at the $r=6$ (geometric units), with the correct infall velocity for small deviations (on the geodesic universal infall), (II) the periapsis advance at large distances, and (III) the presence of a marginally bound circular orbit with specific angular momentum $L=4$, and the periapsis advance of parabolic orbits close to it. We compare the performance of our examples against the Paczyński-Wiita potential and other existing potentials. Finally, we discuss the limitations and advantages of our formulation.
Spectra of standard 1d potentials (double-well, sin-Gordon etc) are given by trans-series in coupling, including (badly divergent) perturbative series (PS), and nonperturbative terms. All of them are badly defined (e.g. PS are badly divergent) but in sum supposed to be good. In this paper we discuss an example of a potential with specially defined couplings making PS completely absent. We calculate its nonperturbative vacuum energy and show that they are reproduced by the action of certain complex solutions to holomorphic Newton equation.
Effective motion planning in high dimensional spaces is a long-standing open problem in robotics. One class of traditional motion planning algorithms corresponds to potential-based motion planning. An advantage of potential based motion planning is composability -- different motion constraints can be easily combined by adding corresponding potentials. However, constructing motion paths from potentials requires solving a global optimization across configuration space potential landscape, which is often prone to local minima. We propose a new approach towards learning potential based motion planning, where we train a neural network to capture and learn an easily optimizable potentials over motion planning trajectories. We illustrate the effectiveness of such approach, significantly outperforming both classical and recent learned motion planning approaches and avoiding issues with local minima. We further illustrate its inherent composability, enabling us to generalize to a multitude of different motion constraints.
We rigorously show that a large family of monotone quantities along the weak inverse mean curvature flow is the limit case of the corresponding ones along the level sets of $p$-capacitary potentials. Such monotone quantities include Willmore and Minkowski-type functionals on Riemannian manifolds with nonnegative Ricci curvature. In $3$-dimensional manifolds with nonnegative scalar curvature, we also recover the monotonicity of the Hawking mass and its nonlinear potential theoretic counterparts. This unified view is built on a refined analysis of $p$-capacitary potentials. We prove that they strongly converge in $W^{1,q}_{\mathrm{loc}}$ as $p\to 1^+$ to the inverse mean curvature flow and their level sets are curvature varifolds. Finally, we also deduce a Gauss-Bonnet-type theorem for level sets of $p$-capacitary potentials.
Zirconia is well-known for plenty of important morphologys with Zr coordination varying from sixfold in the octagonal phase to eightfold in the cubic or tetragonal phase. The development of empirical potentials to describe these zirconia morphologys is an important issue but a long-standing challenge, which becomes a bottleneck for the theoretical investigation of large zirconia structures. In contrast to the standard core-shell model, we develop a new potential for zirconia through the combination of long-range Coulomb interaction and bond order Tersoff model. The bond order characteristic of the Tersoff potential enables it to be well suited for the description of these zirconia morphologys with different coordination numbers. In particular, the complex monoclinic phase with two inequivalent oxygens, that is difficult to be described by most existing empirical potentials, can be well captured by this newly developed potential. It is shown that this potential can provide reasonable predictions for most static and dynamic properties of various zirconia morphologys. Besides its clear physical essence, this potential is at least one order faster than core-shell based potentials in th
This work is dedicated to foundational aspects of general (nonlinear second order) potential theories and fully nonlinear elliptic PDEs. In particular, we systematically develop the fundamental role played by semiconvex functions as a bridge between the classical and viscosity theory of generalized subharmonics determined by a given subequation constraint set in the bundle of $2$-jets over open subsets of Euclidian spaces. The first part is dedicated to four fundamental and complementary aspects of semiconvex analysis for the viscosity theory of subharmonic functions in general potential theories: differentiability properties of first and second order for locally semiconvex functions; a detailed analysis of their upper contact jets and upper contact points; the deep analytical results which ensure the existence of sets of contact points with positive measure; the well-known device of semiconvex approximation of upper semicontinuous functions. The second part is dedicated to general potential theories. The fundamental roles of duality and monotonicity are discussed, along with the main tools in the viscosity theory of subharmonics and the essential role that semiconvex functions pla
We investigate the impact of magnetic fields on the potential barrier between two interacting nuclei. We addressed this by solving the Boltzmann equation and Maxwell's theory in the presence of a magnetic field, resulting in the determination of magnetized permittivity. Additionally, we derived the magnetized Debye potential, which combines the conventional Debye potential with an additional magnetic component. We then compared the Boltzmann approach with the Debye method. Both methods consistently demonstrate that magnetic fields increase permittivity. This enhanced permittivity leads to a reduction in the potential barrier, consequently increasing the reaction rate for nucleosynthesis. Furthermore, the dependence on temperature and electron density in each approach is consistent. Our findings suggest that magnetized plasmas, which have existed since the Big Bang, have played a crucial role in nucleosynthesis.
In theories with spontaneous symmetry breaking, the loop expansion of the effective potential is awkward. In such theories, the exact effective potential $V(φ_c,T)$ is real and convex (as a function of the classical field $φ_c$), but its perturbative series can be complex with a real part that is concave. These flaws limit the utility of the effective potential, particularly in studies of the early universe. A generalization of the effective potential is available that is real, that has no obvious convexity problems, and that can be computed in perturbation theory. For the theory with classical potential $V(φ) = (λ/4)(φ^2 - σ^2)^2$, this more-effective potential closely tracks the usual effective potential where the latter is real $|φ_c| \geq σ/\sqrt{3}$ and naturally extends it to $φ_c = 0$, revealing that the critical temperature at the one-loop level runs from $T_C \approx 1.81 σ$ for $λ= 0.1$ to $T_C \approx 1.74 σ$ for $λ= 1$.
We present a new method for determining the Galactic gravitational potential based on forward modeling of tidal stellar streams. We use this method to test the performance of smooth and static analytic potentials in representing realistic dark matter halos, which have substructure and are continually evolving by accretion. Our FAST-FORWARD method uses a Markov Chain Monte Carlo algorithm to compare, in 6D phase space, an "observed" stream to models created in trial analytic potentials. We analyze a large sample of streams evolved in the Via Lactea II (VL2) simulation, which represents a realistic Galactic halo potential. The recovered potential parameters are in agreement with the best fit to the global, present-day VL2 potential. However, merely assuming an analytic potential limits the dark matter halo mass measurement to an accuracy of 5 to 20%, depending on the choice of analytic parametrization. Collectively, mass estimates using streams from our sample reach this fundamental limit, but individually they can be highly biased. Individual streams can both under- and overestimate the mass, and the bias is progressively worse for those with smaller perigalacticons, motivating the
We estimate the effective heavy quark and antiquark potential in the quark gluon plasma using the gravity dual theory. Two models are considered: $AdS_{5}$ and Sakai-Sugimoto model. The effective potential, obtained by using the rotating fundamental open-string configurations, has the angular momentum dependence which generalizes the static central potential. Motivated by the asymptotic form for zero angular momentum state, we fit the effective potential for $J=1,2$ states in the binding region. The fitting parameters are found to be functions of temperature. Finally, we discuss the differences and similarities of the effective potential between the two gravity dual models. An interesting result is that position of the minimum of potential is determined only by angular momentum and independent of the temperature.
Granular materials, such as sand or grain, exhibit many structural and dynamic characteristics similar to those observed in molecular systems, despite temperature playing no role in their properties. This has led to an effort to develop a statistical mechanics for granular materials that has focused on establishing an equivalent to the microcanonical ensemble and a temperature-like thermodynamic variable. Here, we expand on these ideas by introducing a granular potential into the Edwards ensemble, as an analogue to the chemical potential, and explore its properties using a simple model of a granular system. A simple kinetic Monte Carlo simulation of the model shows the effect of mass transport leading to equilibrium and how this is connected to the redistribution of volume in the system. An exact analytical treatment of the model shows that the compactivity and the ratio of the granular potential to the compactivity determine the equilibrium between two open systems that are able to exchange volume and particles, and that mass moves from high to low values of this ratio. Analysis of the granular potential shows that adding a particle to the system increases the entropy at high comp
We perform a detailed analysis of the Two-Higgs Doublet Model (2HDM) potential. At the tree-level, the potential may accommodate more than one minima, one of them being the electroweak (EW) minimum where the universe lives. The parameter space allowed after the data from the Large Hadron Collider (LHC) came in almost excludes those cases where the EW vacuum is shallower than the second minimum. We extend the analysis by including terms in the 2HDM potential that break the $Z_2$ symmetry of the potential by dimension-4 operators and show that the conclusions remain unchanged. Furthermore, a one-loop analysis of the potential is performed for both cases, namely, where the $Z_2$ symmetry of the potential is broken by dimension-2 or dimension-4 operators. For quantitative analysis, we show our results for the Type-II 2HDM, qualitative results remaining the same for other 2HDMs. We find that the nature of the vacua from the tree-level analysis does not change; the EW vacuum still remains deeper.
Our notation: Points in $\{0,1\}^{\mathbb{Z}-\{0\}} =\{0,1\}^\mathbb{N}\times \{0,1\}^\mathbb{N}=Ω^{-} \times Ω^{+}$, are denoted by $( y|x) =(...,y_2,y_1|x_1,x_2,...)$, where $(x_1,x_2,...) \in \{0,1\}^\mathbb{N}$, and $(y_1,y_2,...) \in \{0,1\}^\mathbb{N}$. The bijective map $\hatσ(...,y_2,y_1|x_1,x_2,...)= (...,y_2,y_1,x_1|x_2,...)$ is called the bilateral shift and acts on $\{0,1\}^{\mathbb{Z}-\{0\}}$. Given $A: \{0,1\}^\mathbb{N}=Ω^+\to \mathbb{R}$ we express $A$ in the variable $x$, like $A(x)$. In a similar way, given $B: \{0,1\}^\mathbb{N}=Ω^{-}\to \mathbb{R}$ we express $B$ in the variable $y$, like $B(y)$. Finally, given $W: Ω^{-} \times Ω^{+}\to \mathbb{R}$, we express $W$ in the variable $(y|x)$, like $W(y|x)$. By abuse of notation we write $A(y|x)=A(x)$ and $B(y|x)=B(y).$ The probability $μ_A$ denotes the equilibrium probability for $A: \{0,1\}^\mathbb{N}\to \mathbb{R}$. Given a continuous potential $A: Ω^+\to \mathbb{R}$, we say that the continuous potential $A^*: Ω^{-}\to \mathbb{R}$ is the dual potential of $A$, if there exists a continuous $W: Ω^{-} \times Ω^{+}\to \mathbb{R}$, such that, for all $(y|x) \in \{0,1\}^{\mathbb{Z}-\{0\}}$ $$ A^* (y) = \left[ A \circ \h
We investigate the problem of fine tuning of the potential in the KKLMMT warped flux compactification scenario for brane-antibrane inflation in Type IIB string theory. We argue for the importance of an additional parameter psi_0 (approximated as zero by KKLMMT), namely the position of the antibrane, relative to the equilibrium position of the brane in the absence of the antibrane. We show that for a range of values of a particular combination of the Kahler modulus, warp factor, and psi_0, the inflaton potential can be sufficiently flat. We point out a novel mechanism for dynamically achieving flatness within this part of parameter space: the presence of multiple mobile branes can lead to a potential which initially has a metastable local minimum, but gradually becomes flat as some of the branes tunnel out. Eventually the local minimum disappears and the remaining branes slowly roll together, with assisted inflation further enhancing the effective flatness of the potential. With the addition of Kahler and superpotential corrections, this mechanism can completely remove the fine tuning problem of brane inflation, within large regions of parameter space. The model can be falsified if
An inverse problem for the two-dimensional Schrodinger equation with $L^p_{com}$-potential, $p>1$, is considered. Using the $\overline{\partial}$-method, the potential is recovered from the Dirichlet-to-Neumann map on the boundary of a domain containing the support of the potential. We do not assume that the potential is small or that the Faddeev scattering problem does not have exceptional points. The paper contains a new estimate on the Faddeev Green function that immediately implies the absence of exceptional points near the origin and infinity when $v\in L^p_{com}$.
In this paper we bring together some of the key ideas and methods of two disparate fields of mathematical research, frame theory and optimal transport, using the methods of the second to answer questions posed in the first. In particular, we construct gradient flows in the Wasserstein space $P_2(\mathbb{R}^d)$ for a new potential, the tightness potential, which is a modification of the probabilistic frame potential. It is shown that the potential is suited for the application of a gradient descent scheme from optimal transport that can be used as the basis of an algorithm to evolve an existing frame toward a tight probabilistic frame.
We describe a method of obtaining the inflationary potential from observations which does not use the slow-roll approximation. Rather, the microwave anisotropy spectrum is obtained directly from a parametrized potential numerically, with no approximation beyond linear perturbation theory. This permits unbiased estimation of the parameters describing the potential, as well as providing the full error covariance matrix. We illustrate the typical uncertainties obtained using the Fisher information matrix technique, studying the $λφ^4$ potential in detail as a concrete example.
Singularity of the potential function makes quantum tunneling problem mathematically underdetermined. To circumvent the difficulties it introduced in physics, a potential singularity cutoff is often used, followed by a reverse limit transition, or is a suitable self-adjoint extension of the Hamiltonian along the entire coordinate axis made. However, both of them somehow affect the singular nature of the problem, and so I discuss here how quantum tunneling will behave if the original singular nature of the Schrodinger equation left untuched. To do this, I use the property of the probability density current that the singularities are mutually destroyed in it. It is found that the mildly singular potential has a finite, but unusual tunneling transparency, in particular, a non-zero value at zero energy of incident particle. The tunneling of one dimensional Coulomb potential exhibits infinitely fast and complete oscillation at the zero energy boundary and a suppresion to zero in the high-energy limit. In the more singular region, the tunneling becomes forbidden, theby repeating the well-known result of the regularized counterparts.
In this study a computational method of the multi-reference VCA(virtual crystal approximation) pseudo-potential generation is presented. This is an extension of that proposed Ramer and Rappe [J. Phys. Chem. Sol. 61, 315(2000)], the scheme of which is in want of the explicit incorporation of semi-core states. To compensate this drawback, a kind of fine tuning applied to the non-multi-reference VCA pseudo-potential; the form of the pseudo-potential is slightly modified within the cut-off radius in order that the agreements between the pseudo-potential and all-electron calculations are guaranteed both for semi-core and valence states. The improvement in the present work is validated by atomic and crystalline test calculations for the transferability and the lattice constant estimation.