Search Results (59)

The Feynman path integral formalism for non-relativistic quantum mechanics is revisited. A comparison is made with cases of light propagation (Huygens’ principle) and Brownian motion. The difficulties for a physical model applying Feynman’s formalism are pointed out. A reformulation is proposed, where the transition probability of a particle from one space-time point to another one is the sum of probabilities of the possible paths. As an application, Born approximation for scattering is derived within the formalism, which suggests an interpretation involving the stochastic motion of a particle rather than the square of a wavelike amplitude. Full article

During the last few decades, several approximate, but useful, methods for including dynamical quantum effects in molecular simulations have been developed. These methods can be applied to systems with hundreds of degrees of freedom and with arbitrary potentials. Among these methods, we find the Feynman–Kleinert linearized path integral model, including its planetary versions, which are the focus of this review. The aim is to calculate quantum correlation functions for complex systems. Many important properties, e.g., transport coefficients, may thus be obtained. We summarize important applications of the method, and compare them to alternative ones, such as centroid molecular dynamics and ring polymer molecular dynamics. We finally discuss possible future improvements of the FK-LPI method. Full article

This research introduces an innovative probabilistic method for examining torsional stress behavior in spherical shell structures through Monte Carlo simulation techniques. The spherical geometry of these components creates distinctive computational difficulties for conventional analytical and deterministic numerical approaches when solving torsion-related problems. The authors develop a comprehensive mesh-free Monte Carlo framework built upon the Feynman–Kac formula, which maintains the geometric symmetry of the domain while offering a probabilistic solution representation via stochastic processes on spherical surfaces. The technique models Brownian motion paths on spherical surfaces using the Euler–Maruyama numerical scheme, converting the Saint-Venant torsion equation into a problem of stochastic integration. The computational implementation utilizes the Fibonacci sphere technique for achieving uniform point placement, employs adaptive time-stepping strategies to address pole singularities, and incorporates efficient algorithms for boundary identification. This symmetry-maintaining approach circumvents the mesh generation complications inherent in finite element and finite difference techniques, which typically compromise the problem’s natural symmetry, while delivering comparable precision. Performance evaluations reveal nearly linear parallel computational scaling across up to eight processing cores with efficiency rates above 70%, making the method well-suited for multi-core computational platforms. The approach demonstrates particular effectiveness in analyzing torsional stress patterns in thin-walled spherical components under both symmetric and asymmetric boundary scenarios, where traditional grid-based methods encounter discretization and convergence difficulties. The findings offer valuable practical recommendations for material specification and structural design enhancement, especially relevant for pressure vessel and dome structure applications experiencing torsional loads. However, the probabilistic characteristics of the method create statistical uncertainty that requires cautious result interpretation, and computational expenses may surpass those of deterministic approaches for less complex geometries. Engineering analysis of the outcomes provides actionable recommendations for optimizing material utilization and maintaining structural reliability under torsional loading conditions. Full article

Viruses that infect bacteria (bacteriophages or phages) have a history of use in both biomedicine and basic molecular biology. Here, I briefly outline the pre-1940 use of phages in biomedicine and then more comprehensively outline the subsequent use of phages in determining the basics of molecular biology. Finally, I outline work that appears to form the foundation for a future, phage-enhanced biomedicine that generally extends medicine in the areas of anti-bacterial therapy (including vaccinology), anti-tumor therapy, and understanding the basic process of amyloid-associated neurodegenerative diseases. The following are general conclusions. (1) In the future, the discipline of phage-based biomedicine will be enhanced by more extensive merging with the discipline of basic phage biology (including molecular biology) and evolution. These two disciplines have been separated post-1940. (2) Biomedicine, in general, will be assisted if the focus is on key problems and key observations, thereby leaving details to later work. (3) Simplicity of strategy is a virtue that can be implemented and should be pursued with phages. (4) Capacity for directed evolution provides phages with generative (artificial intelligence-like) means for increasing biomedical effectiveness without using human design. Two related quotes set the stage (references at the end of the text). “But see that the imagination of nature is far, far greater than the imagination of man” (physicist Richard Feynman). “Nature, in all its variations and seeming paradoxes, speaks to those who pay attention and gives hints and clues to basic facts” (a thought attributed to Felix d’Herelle, a self-trained biologist who developed biological phage isolation and characterization). The integration of natural phenomenon-focused basic science and medical practice is an underlying theme. Full article

In this study, we present a novel mathematical framework for pricing financial derivates and modelling asset behaviour by bringing together fractional Brownian motion (fBm), fuzzy logic, and jump processes, all aligned with no-arbitrage principle. In particular, our mathematical developments include fBm defined through Mandelbrot-Van Ness kernels, and advanced mathematical tools such Molchan martingale and BDG inequalities ensuring rigorous theoretical validity. We bring together these different concepts to model uncertainties like sudden market shocks and investor sentiment, providing a fresh perspective in financial mathematics and derivatives pricing. By using fuzzy logic, we incorporate subject factors such as market optimism or pessimism, adjusting volatility dynamically according to the current market environment. Fractal mathematics with the Hurst exponent close to zero reflecting rough market conditions and fuzzy set theory are combined with jumps, representing sudden market changes to capture more realistic asset price movements. We also bridge the gap between complex stochastic equations and solvable differential equations using tools like Feynman-Kac approach and Girsanov transformation. We present simulations illustrating plausible scenarios ranging from pessimistic to optimistic to demonstrate how this model can behave in practice, highlighting potential advantages over classical models like the Merton jump diffusion and Black-Scholes. Overall, our proposed model represents an advancement in mathematical finance by integrating fractional stochastic processes with fuzzy set theory, thus revealing new perspectives on derivative pricing and risk-free valuation in uncertain environments. Full article

In this work, we analyze the vanishing cycles of Feynman loop integrals by the means of the Mayer–Vietoris spectral sequence. A complete classification of possible vanishing geometries is obtained. We use this result for establishing an asymptotic expansion for the loop integrals near their singularity locus and then give explicit formulas for the coefficients of such an expansion. Further development of this framework may potentially lead to exact calculations of one- and two-loop Feynman diagrams, as well as other next-to-leading and higher-order diagrams, in studies of radiative corrections for upcoming lepton–hadron scattering experiments. Full article

This paper defines stubbornness as an optimal feedback Nash equilibrium within a dynamic setting. Stubbornness is treated as a player-specific parameter, with the team’s coach initially selecting players based on their stubbornness and making substitutions during the game according to this trait. The payoff function of a soccer player is evaluated based on factors such as injury risk, assist rate, pass accuracy, and dribbling ability. Each player aims to maximize their payoff by selecting an optimal level of stubbornness that ensures their selection by the coach. The goal dynamics are modeled using a backward parabolic partial stochastic differential equation (BPPSDE), leveraging its theoretical connection to the Feynman–Kac formula, which links stochastic differential equations (SDEs) to partial differential equations (PDEs). A stochastic Lagrangian framework is developed, and a path integral control method is employed to derive the optimal measure of stubbornness. The paper further applies a variant of the Ornstein–Uhlenbeck BPPSDE to obtain an explicit solution for the player’s optimal stubbornness. Full article

Many variations of the multiple zeta values have been found to play important roles in different branches of mathematics and theoretical physics in recent years, such as the cyclotomic/color version, which appears prominently in the computation of Feynman integrals. In this paper, we introduce a trigonometric variant of the Kaneko–Tsumura $\psi$-function (called the Kaneko–Tsumura $\tilde{\psi }$-function) and discover some nice properties similar to those for ordinary Kaneko–Tsumura $\psi$-values using the method of iterated integrals, which was first studied systematically by K.T. Chen in the 1960s. In particular, we establish some duality formulas involving the Kaneko–Tsumura $\tilde{\psi }$-function and alternating multiple T-values by adapting Yamamoto’s graphical representation method for computing special types of iterated integrals. Full article

The current developments in the theory of quantum static triplet correlations and their associated structures (real r-space and Fourier k-space) in monatomic fluids are reviewed. The main framework utilized is Feynman’s path integral formalism (PI), and the issues addressed cover quantum diffraction effects and zero-spin bosonic exchange. The structures are associated with the external weak fields that reveal their nature, and due attention is paid to the underlying pair-level structures. Without the pair, level one cannot fully grasp the triplet extensions in the hierarchical ladder of structures, as both the pair and the triplet structures are essential ingredients in the triplet response functions. Three general classes of PI structures do arise: centroid, total continuous linear response, and instantaneous. Use of functional differentiation techniques is widely made, and, as a bonus, this leads to the identification of an exact extension of the “classical isomorphism” when the centroid structures are considered. In this connection, the direct correlation functions, as borrowed from classical statistical mechanics, play a key role (either exact or approximate) in the corresponding quantum applications. Additionally, as an auxiliary framework, the traditional closure schemes for triplets are also discussed, owing to their potential usefulness for rationalizing PI triplet results. To illustrate some basic concepts, new numerical calculations (path integral Monte Carlo PIMC and closures) are reported. They are focused on the purely diffraction regime and deal with supercritical helium-3 and the quantum hard-sphere fluid. Full article

In a recent contribution, we identified possible points of contact between the asymptotically safe and canonical approaches to quantum gravity. The idea is to start from the reduced phase space (often called relational) formulation of canonical quantum gravity, which provides a reduced (or physical) Hamiltonian for the true (observable) degrees of freedom. The resulting reduced phase space is then canonically quantized, and one can construct the generating functional of time-ordered Wightman (i.e., Feynman) or Schwinger distributions, respectively, from the corresponding time-translation unitary group or contraction semigroup, respectively, as a path integral. For the unitary choice, that path integral can be rewritten in terms of the Lorentzian Einstein–Hilbert action plus observable matter action and a ghost action. The ghost action depends on the Hilbert space representation chosen for the canonical quantization and a reduction term that encodes the reduction of the full phase space to the phase space of observables. This path integral can then be treated with the methods of asymptotically safe quantum gravity in its Lorentzian version. We also exemplified the procedure using a concrete, minimalistic example, namely Einstein–Klein–Gordon theory, with as many neutral and massless scalar fields as there are spacetime dimensions. However, no explicit calculations were performed. In this paper, we fill in the missing steps. Particular care is needed due to the necessary switch to Lorentzian signature, which has a strong impact on the convergence of “heat” kernel time integrals in the heat kernel expansion of the trace involved in the Wetterich equation and which requires different cut-off functions than in the Euclidian version. As usual we truncate at relatively low order and derive and solve the resulting flow equations in that approximation. Full article

This paper presents a novel approach to modeling the repertoire of the immune system and its adaptation in response to the evolutionary dynamics of pathogens associated with their genetic variability. It is based on application of a dynamic programming-based framework to model the antigen-driven immune repertoire synthesis. The processes of formation of new receptor specificity of lymphocytes (the growth of their affinity during maturation) are described by an ordinary differential equation (ODE) with a piecewise-constant right-hand side. Optimal control synthesis is based on the solution of the Hamilton–Jacobi–Bellman equation implementing the dynamic programming approach for controlling Gaussian random processes generated by a stochastic differential equation (SDE) with the noise in the form of the Wiener process. The proposed description of the clonal repertoire of the immune system allows us to introduce an integral characteristic of the immune repertoire completeness or the integrative fitness of the whole immune system. The quantitative index for characterizing the immune system fitness is analytically derived using the Feynman–Kac–Kolmogorov equation. Full article

This paper presents a novel approach to identify an optimal coefficient for evaluating a player’s batting average, strike rate, and bowling average, aimed at achieving an optimal team score through dynamic modeling using a path integral method. Additionally, it introduces a new model for run dynamics, represented as a stochastic differential equation, which factors in the average weather conditions at the cricket ground, the specific weather conditions on the match day (including abrupt changes that may halt the game), total attendance, and home field advantage. An analysis of real data is been performed to validate the theoretical results. Full article

Cortical neurons integrate upstream signals and random electrical noise to gate signaling outcomes, leading to statistically random patterns of activity. Yet classically, the neuron is modeled as a binary computational unit, encoding Shannon entropy. Here, the neuronal membrane potential is modeled as a function of inherently probabilistic ion behavior. In this new model, each neuron computes the probability of transitioning from an off-state to an on-state, thereby encoding von Neumann entropy. Component pure states are integrated into a physical quantity of information, and the derivative of this high-dimensional probability distribution yields eigenvalues across the multi-scale quantum system. In accordance with the Hellman–Feynman theorem, the resolution of the system state is paired with a spontaneous shift in charge distribution, so this defined system state instantly becomes the past as a new probability distribution emerges. This mechanistic model produces testable predictions regarding the wavelength of free energy released upon information compression and the temporal relationship of these events to physiological outcomes. Overall, this model demonstrates how cortical neurons might achieve non-deterministic signaling outcomes through a computational process of noisy coincidence detection. Full article

In this paper, we consider and study in detail the generalized Fox–Wright function ${}_p\tilde{\Psi }_q$ introduced in our recent work as an extension of the Fox–Wright function ${}_p\Psi _q$. This special function can be seen as an important case of the so-called I-functions of Rathie and $\overline{H}$-functions of Inayat-Hussain, that in turn extend the Fox H-functions and appear to include some Feynman integrals in statistical physics, in polylogarithms, in Riemann Zeta-type functions and in other important mathematical functions. Depending on the parameters, ${}_p\tilde{\Psi }_q$ is an entire function or is analytic in an open disc with a final radius. We derive its basic properties, such as its order and type, and its images under the Laplace transform and under classical fractional-order integrals. Particular cases of ${}_p\tilde{\Psi }_q$ are specified, including the Mittag-Leffler and Le Roy-type functions and their multi-index analogues and many other special functions of Fractional Calculus. The corresponding results are illustrated. Finally, we emphasize the role of these new generalized hypergeometric functions as eigenfunctions of operators of new Fractional Calculus with specific I-functions as singular kernels. This paper can be considered as a natural supplement to our previous surveys “Going Next after ‘A Guide to Special Functions in Fractional Calculus’: A Discussion Survey”, and “A Guide to Special Functions of Fractional Calculus”, published recently in this journal. Full article

A Silicon Photonics modulator is a high-speed photonic integrated circuit for optical data transmission in high-capacity optical networks. Silicon Photonics modulators in the configuration of a Mach–Zehnder interferometer, in which a PN-junction rib-waveguide phase shifter is inserted in each arm of the interferometer, are studied in this paper because of their superior performance of high-quality optical data generation in a wide range of spectral bands and their simplicity in fabrication processes suitable to production in foundries. Design, fabrication, and fundamental characteristics of Silicon Photonics Mach–Zehnder modulators are reviewed as an introduction to these high-speed PICs on the Silicon Photonics platform. Modulation speed, or modulation bandwidth, is a key performance item, as well as optical loss, in the application to high-speed optical transmitters. Limiting factors on modulation speed are addressed in equations. Electrical resistance–capacitance coupling, which causes optical modulation bandwidth–optical loss trade-off, is the most challenging limiting factor that limits high-speed modulation. Expansion of modulation bandwidth is not possible without increasing optical loss in the conventional approaches. A new idea including quantum-mechanical effect in the design of Silicon Photonics modulators is proposed and proved in computational analysis to resolve the bandwidth loss trade-off. By adding high-mobility quantum-well overlayers to the side slab wings of the rib-waveguide phase shifter, the modulation bandwidth is doubled without increasing optical loss to achieve a 200 Gbaud modulation rate. Full article