Exact Analytical Expressions Research Articles

Nonreciprocal forces enable cold-to-hot heat transfer between nanoparticles

We study the heat transfer between two nanoparticles held at different temperatures that interact through nonreciprocal forces, by combining molecular dynamics simulations with stochastic thermodynamics. Our simulations reveal that it is possible to construct nano refrigerators that generate a net heat transfer from a cold to a hot reservoir at the expense of power exerted by the nonreciprocal forces. Applying concepts from stochastic thermodynamics to a minimal underdamped Langevin model, we derive exact analytical expressions predictions for the fluctuations of work, heat, and efficiency, which reproduce thermodynamic quantities extracted from the molecular dynamics simulations. The theory only involves a single unknown parameter, namely an effective friction coefficient, which we estimate fitting the results of the molecular dynamics simulation to our theoretical predictions. Using this framework, we also establish design principles which identify the minimal amount of entropy production that is needed to achieve a certain amount of uncertainty in the power fluctuations of our nano refrigerator. Taken together, our results shed light on how the direction and fluctuations of heat flows in natural and artificial nano machines can be accurately quantified and controlled by using nonreciprocal forces.

Simulated diffusion spreadability for characterizing the structure and transport properties of two-phase materials

Time-dependent diffusion processes between phases of heterogeneous materials are ubiquitous in a variety of contexts in the physical, chemical, and biological sciences. Examples of such materials include composites, porous materials, geological media, cellular solids, polymer blends, colloids, gels, and biological media. The recently developed diffusion spreadability, S(t), provides a direct link between time-dependent interphase diffusive transport and the microstructure of two-phase materials across length scales (Torquato, 2021); thus making S(t) a powerful dynamic means for classifying all statistically homogeneous microstructures, spanning from anti-hyperuniform to hyperuniform. It was shown that the small-, intermediate-, and long-time behaviors of S(t) are directly determined by the small-, intermediate-, and large-scale structural features of the material. Moreover, the spreadability can be applied as a physical-property based tool for microstructural characterization in the absence of or as supplement to scattering information. In this work, we develop a computationally efficient algorithm for ascertaining S(t) directly from digitized representations of material microstructures via random-walk techniques. Our algorithm yields the time-dependent local walker concentration field c(x,t), a quantity not previously examined in the context of the spreadability, enabling us to compute the entropy production rate ṡ(t) of the associated diffusion process, which is a quantity related to the rate of energy dissipation. We also derive exact analytical expressions for ṡ(t), and find that hyperuniform materials have smaller dissipation than any nonhyperuniform materials. Lastly, we use our algorithm to compute, for the first time, the more general case of the spreadability in which the phase diffusion coefficients are distinct and provide a method for extracting the effective diffusion coefficient of the two-phase material from such data. We apply our algorithm to a variety of two- and three-dimensional simulated model (non)hyperuniform microstructures to assess their large-scale structures and diffusion properties. Given previously identified connections between the spreadability and certain nuclear magnetic resonance (NMR) and magnetic resonance imaging (MRI) procedures, our algorithm is also pertinent to such experimental studies. Overall, our algorithm has practical use in the discovery and design of heterogeneous materials with desirable time-dependent diffusion properties.

Frustration and ordering in Ising chain in an external magnetic field with third-neighbor interactions

In this paper, the frustration properties of the Ising model on a one-dimensional monoatomic equidistant lattice in an external magnetic field are investigated, taking into account the exchange interactions of atomic spins at the sites of the first, second, and third neighbors. Exact analytical expressions for the thermodynamic functions of the system are obtained by the Kramers–Wannier transfer-matrix method. A magnetic phase diagram of the ground state of such a spin system is constructed and studied thoroughly. The points and lines of frustrations of the system depending on the values and signs of exchange interactions and on an external magnetic field are found. The criteria for the occurrence of magnetic frustrations in the presence of competition between the energies of exchange interactions and an external magnetic field are formulated. The peculiar features are investigated and the values of entropy and magnetization of the ground state of this model are obtained in the frustration regime and beyond it. Various types of behavior of entropy, magnetization, and magnetic susceptibility depending on the model parameters are revealed.

Electrical property of an m × n apple surface network

The circuit network model has important application value in many disciplines, and many complex problems can be simulated by establishing the network model. However, looking for the explicit solutions of the electrical property in an arbitrary resistor network is important but difficult. In this paper an arbitrary m × n apple surface network is researched by the Recursion-Transform method with voltage parameters (RT-V) for the first time, and the RT-V method consists of four rigorous basic steps. We study and derive the exact analytical expression of potential function and the equivalent resistance formula, and some special cases and applications of them are also discussed. The theoretical results obtained are concise and provide a new theoretical basis for the research of related subjects.

Interplay between Non-Markovianity of Noise and Dynamics in Quantum Systems.

The non-Markovianity of open quantum system dynamics is often associated with the bidirectional interchange of information between the system and its environment, and it is thought to be a resource for various quantum information tasks. We have investigated the non-Markovianity of the dynamics of a two-state system driven by continuous time random walk-type noise, which can be Markovian or non-Markovian depending on its residence time distribution parameters. Exact analytical expressions for the distinguishability as well as the trace distance and entropy-based non-Markovianity measures are obtained and used to investigate the interplay between the non-Markovianity of the noise and that of dynamics. Our results show that, in many cases, the dynamics are also non-Markovian when the noise is non-Markovian. However, it is possible for Markovian noise to cause non-Markovian dynamics and for non-Markovian noise to cause Markovian dynamics but only for certain parameter values.

Topological, Spectroscopic and Energetic Properties of Cycloparaphenylene Series

Carbon nanorings are macrocyclic aromatic hydrocarbons that represent the segments of carbon nanotubes and have received considerable attention due to their novel nanoelectronic and photophysical capabilities. The cycloparaphenylene (CPP), a carbon nanoring with a ring-like structure comprising of only benzene units, has been the subject matter of several studies. The carbon nanoring structures bearing polycyclic aromatic hydrocarbons such as hexabenzocoronene and pyrene molecules are also found to exhibit exciting characteristics and applications. This in turn evolves a wide range of novel chemical structures whose behavior and attributes are to be studied and explored. Molecular descriptors help us to study such novel nanostructures by providing a greater understanding of their unique properties, whereas such characterization remains challenging due to their ring-shaped arrangement. In this study, we investigate cycloparaphenylene series and show that such structures are not bounded by the partial cube family. We also derive the exact analytic expressions of degree, distance and closeness-related descriptors. We have also computed the thermodynamic and kinetic stabilities of the three structures using the graph spectra, resonance energies, HOMO-LUMO gaps and Kekulé counts obtained from the graph spectra and combinatorial methods. Whereas the CPPhbc[5] structure is thermodynamically more stable, the CPPyr[9] is kinetically more stable. Furthermore, we have employed combinatorial generating function methods to compute the NMR and ESR spectra of the title compounds.

The trapping problem on horizontal partitioned level-3 sierpinski gasket networks

Random walk on complex networks is a research hotspot nowadays. The average trapping time (ATT) is an important property related to the trapping problem, which is a variant of random walk, because it can be used to measure the transmission efficiency of particles randomly walking on the network. In this paper, we consider the trapping problem on the horizontal partitioned level-3 Sierpinski gasket network which is determined by the cutting line l k , that is, by the partition coefficient k. Then through the structure of this research model, we derive the exact analytical expression of the ATT. Furthermore, we draw two kinds of numerical simulation diagrams to simulate the relationship between the ATT and the iteration number and the partition coefficient, and compare them with the ATT on the original graph (uncut). The obtained solution shows that the ATT is affected by the k, specifically, the larger the k, the shorter the ATT, that is the higher the transmission efficiency.

Full analytical solution of finite-length armchair/zigzag nanoribbons

Finite-length armchair graphene nanoribbons can behave as one-dimensional topological materials, that may show edge states in their zigzag-terminated edges, depending on their width and termination. We show here a full solution of tight-binding graphene rectangles of any length and width that can be seen as either finite-length armchair or zigzag ribbons. We find exact analytical expressions for both bulk and edge eigenstates and eigenenergies. We write down exact expressions for the Coulomb interactions among edge states and introduce a Hubbard-dimer model to analyze the emergence and features of different magnetic states at the edges, whose existence depends on the ribbon length. We find ample room for experimental testing of our predictions in $N=5$ armchair ribbons. We compare the analytical results with ab initio simulations to benchmark the quality of the dimer model and to set its parameters. A further detailed analysis of the ab initio Hamiltonian allows us to identify those variations of the tight-binding parameters that affect the topological properties of the ribbons.

Bending-bending coupled static analysis of functionally graded and porous pretwisted cantilever beams using initial values method

In this study, the bending-bending coupled static behavior of functionally graded and porous pretwisted cantilever Timoshenko beams is reported. Symmetric power-law distribution is employed as the material profile and it is assumed that porosity can disperse over the cross-section in two forms namely, even and uneven distributions. Two different yet equivalent forms of governing equations are obtained by formulating the problem in the fixed and curvilinear coordinate systems, respectively. The initial values method is adopted as the solution methodology. While the fundamental matrix is computed numerically by using the multiplicative integral of Volterra to obtain the solution in the fixed coordinate formulation, the matrix exponential which allows for obtaining exact expressions is utilized in the curvilinear coordinate formulation. Exact analytical deflection expressions are presented for space-fixed vertical tip force and body-fixed uniform distributed force cases. It is shown that both formulation approaches yield consistent results with each other and those obtained from the finite element analyses. In addition, several parametric studies that investigate the effects of the volume fraction index, porosity coefficient, and porosity distribution on the tip deflections are presented.

Photon Recycling in a Solar Cell With Two Lambertian Surfaces

An exact analytical expression for the probability of photon reabsorption and recycling in an idealized solar cell with two Lambertian surfaces is derived. The existing approximations are found to agree with the exact formula to within 5%. The most accurate approximation turned out to be the simplest one that sets the reabsorption probability to the weak-absorption limit of the cell absorptance. The maximal photoconversion efficiency of a silicon solar cell is evaluated to be 29.5% at the base thickness of 98 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mu$</tex-math></inline-formula> m.

Secrecy Rate of TAS in AF-Relay Systems With Multi-Antenna Relay and User Over Non-Identical Channel Estimation Error

This work evaluates the secrecy rate of transmit antenna selection (TAS) in the amplify and forward (AF)-relay system with multi-antenna relay and user. This paper considers that the receiver of relays, users, and an eavesdropper have non-identical channel estimation error. To assess the secrecy rate, this work derives new analytical expressions for the cumulative distribution function (CDF) and probability density function (PDF) of the received signal to noise ratio (SNR) using infinite series for maximal-ratio combining (MRC) and selection combining (SC) receivers. Using the CDF for the received SNR, we derive new exact analytical expressions for the secrecy rate with non-identical channel estimation error for four scenarios. Further, using the PDF for the received SNR, we derive the asymptotic analytical expressions for the secrecy rate and quantify the high SNR slope and the high SNR power offset for four scenarios. From the numerical results, we observe that the relay and user selection has the high SNR power offset gain compared to the non-selection. The secrecy rate of the proposed system can be improved according to legitimate receiver's parameters such as the number of receive antennas and channel correlation coefficient and can be degraded according to eavesdropper's parameters. The superiority of the secrecy rate of the MRC receiver and the SC receiver depends on the channel estimation accuracy of the MRC receiver.

Topological Indices, Graph Spectra, Entropies, Laplacians, and Matching Polynomials of n-Dimensional Hypercubes

We obtain a large number of degree and distance-based topological indices, graph and Laplacian spectra and the corresponding polynomials, entropies and matching polynomials of n-dimensional hypercubes through the use of Hadamard symmetry and recursive dynamic computational techniques. Moreover, computations are used to provide independent numerical values for the topological indices of the 11- and 12-cubes. We invoke symmetry-based recursive Hadamard transforms to obtain the graph and Laplacian spectra of nD-hypercubes and the computed numerical results are constructed for up to 23-dimensional hypercubes. The symmetries of these hypercubes constitute the hyperoctahedral wreath product groups which also pave the way for the symmetry-based elegant computations. These results are used to independently validate the exact analytical expressions that we have obtained for the topological indices as well as graph, Laplacian spectra and their polynomials. We invoke a robust dynamic programming technique to handle the computationally intensive generation of matching polynomials of hypercubes and compute all matching polynomials up to the 6-cube. The distance degree sequence vectors have been obtained numerically for up to 108-dimensional cubes and their frequencies are found to be in binomial distributions akin to the spectra of n-cubes.

Degree-Sum Based Topological Indices of Supercoronene and Triangle-Shaped Discotic Graphene Using NM-Polynomial

The structural features of polycyclic aromatic compounds, like the fundamental molecular topology, strongly influence their chemical and biological properties. Quantitative structure-activity and property relationship (QSAR/QSPR) techniques for predicting characteristics of polycyclic aromatic compounds (PAC) and related graphs from chemical structures have been developed in this approach. To obtain degree-based topological indices, we have many polynomials. The neighborhood M-polynomial is one of these polynomials used to produce several topological indices based on neighborhood degree sum. In this study, we offer the exact analytical expressions of neighborhood M-polynomial and their corresponding topological indices for supercoronene (SC), cove-hexabenzocoronene (cHBC), and triangular-shaped discotic graphene (TDG) with hexabenzocorenene (HBC) as the base molecule. The findings could help with the development of characteristic physicochemical prediction.

Exact Expressions for Lightning Electromagnetic Fields: Application to the Rusck Field-to-Transmission Line Coupling Model

An exact analytical expression for the electric field of the return stroke as excited by a propagating step current source is derived in this paper. This expression could be advantageously used to evaluate the disturbances caused by lightning on overhead lines. There are three equivalent procedures to evaluate the voltages induced by lightning on power lines, namely, the Agrawal–Price–Gurbaxani model, the Taylor–Satterwhite–Harrison model, and the Rachidi model. In the case of a vertical return stroke channel, the coupling model developed by Rusck becomes identical to these three coupling models. Due to its simplicity, the Rusck model is frequently used by engineers to study the effects of lightning on power distribution and transmission lines. In order to reduce the time involved in the electromagnetic field calculation, the Rusck model is incorporated with an analytical expression for the electromagnetic fields of the return stroke excited by a propagating step current pulse. Our research work shows that the Rusck expression can be used to calculate the peak values of lightning induced voltages to an accuracy of about 10%. However, the use of this analytical expression to calculate the time derivatives of lightning induced voltages may result in errors as large as 50%. The derived expression in this paper can be used to correct for this inaccuracy. We also provide an exact expression for the electric field at any given point in space when the propagating current is an impulse function. This expression can be combined with the convolution integral to obtain the electric field corresponding to waveforms similar to measured return stroke currents.

Characterizing superradiant dynamics in atomic arrays via a cumulant expansion approach

Ordered atomic arrays with subwavelength lattice spacing emit light collectively. For fully inverted atomic arrays, this results in an initial burst of radiation and a fast buildup of coherences between the atoms at initial times. Based on a cumulant expansion of the equations of motion, we derive exact analytical expressions for the emission properties and numerically analyze the full many-body problem resulting in the collective decay process for unprecedented system sizes of up to a few hundred atoms. We benchmark the cumulant expansion approach and show that it correctly captures the cooperative dynamics resulting in superradiance. For fully inverted arrays, this allows us to extract the scaling of the superradiant peak with particle number. For partially excited arrays where no coherences are shared among atoms, we also determine the critical number of excitations required for the emergence of superradiance in one- and two-dimensional geometries. In addition, we study the robustness of superradiance in the case of non-unit filling and position disorder.

First-principles theory of electrochemical capacitance

The differential capacitance comprises the most relevant thermodynamic information about an electrochemical system. Classical approaches to describe electrochemical capacitance have difficulties to combine the treatment of the ionic contribution of the electrolyte with the electronic contribution of the electrode. Moreover, different approaches are typically required for the description of the double-layer capacitance, on the one hand, and the pseudocapacitive contribution due to adsorption or intercalation of reactive species, on the other. In the present work, a new approach to describe electrochemical capacitance from first principles is developed. The treatment of a general electrochemical system at the level of multicomponent density-functional theory (MCDFT) yields an exact analytical expression for the total capacitance of the system, which corresponds to a formal series-circuit partitioning into “quantum” capacitance contributions of the density of states of electrons and ions, the (mean-field) electrostatic capacitance, as well as capacitance contributions of exchange and correlation among all active species. It is shown that the classical expression of the double-layer capacitance involving the interfacial Galvani potential is obtained in the limit of extended electrode and electrolyte regions. Importantly, the present formalism also describes systems with confined electrode and electrolyte phases, where the definition of the (classical) inner potentials of the electrode and electrolyte domains becomes problematic. Moreover, the new approach unifies the treatment of double-layer capacitance and pseudocapacitance resulting from reactive processes.

Thermal rate of transmission through a barrier: Exact expansion of up to and including terms of order ℏ4

Nine decades after Wigner's formulation of quantum rate theory, his celebrated result was recently generalized to the asymmetric barrier using an exact first-order expansion of the transmission probability in terms of ${\ensuremath{\hbar}}^{2}$. This paper extends the first-order quantum correction to second-order correction of order ${\ensuremath{\hbar}}^{4}$ for the thermally averaged transmission probability through an arbitrary barrier. The derivation employs a systematic expansion of the projection operator onto products and the thermal distribution which involves a Taylor expansion of the potential about the barrier up to eighth order. The resulting exact analytical expression is calibrated with numerical calculations of several model potentials and shows excellent agreement when the ${\ensuremath{\hbar}}^{4}$ term is included. In comparison, the semiclassical transition state theory cannot reproduce the correct ${\ensuremath{\hbar}}^{4}$ terms when the anharmonicity is treated on the level of VPT-4 (vibrational perturbation theory---fourth order) and will potentially need a VPT-6 expansion. Further analysis of the quartic barrier reveals suppressed transmission due to the dominant role of quantum reflection above the barrier. These results not only provide a conceptual framework but can also be applied to heavy atom tunneling and machine learning.

Exact magnetic properties of two-dimensional spin-1 Cooper pairs

Exact magnetic properties of two-dimensional (2D) spin-1 Cooper pairs are investigated within the framework of quantum statistical mechanics. We derive an exact analytical expression of the magnetization of 2D spin-1 Cooper pairs, which involves a q-digamma function and a q-Pochhammer symbol in mathematics. The magnetization density m(T) is a function of temperature T, Landé factor g, number density σ, and magnetic field B. Firstly, we examine the evolution of the magnetization density m(T) with the Landé factor g. m(T) is negative when g < 1, and changes gradually to positive after g c . Here, is the critical value of the Landé factor g, at which . Secondly, we examine the evolution of the magnetization density m(T) with the temperature T. When g < 1, there are the two asymptotic lines for the magnetization density m(T). The low-temperature asymptotic line is nC s−1 and the high-temperature asymptotic line is . Finally, we examine the evolution of the magnetization density m(T) with the number density σ. No matter how large the Landé factor g is, there is a lower cutoff σ c of the number density σ. The lower cutoff number density is about cm−2. If , the Cooper pair system does not support a solution of the reduced chemical potential x. As the number density σ is increased, the Bose–Einstein condensation of 2D Cooper pairs is strengthened.

Unified low-order modeling approach to can-annular combustors

We present a unified, physics-based low-order modeling approach to can-annular combustors, combining a linear stability analysis and a nonlinear Langevin approach to unravel the highly complex, multi-physical dynamics in this type of system. In the linear part, Howe's Rayleigh conductivity model is combined with the projected Helmholtz equation and a Bloch wave ansatz to arrive at a single equation for the frequency spectrum of an N-can combustor. Starting from first principles, we illustrate and give a physical explanation for the coupling-induced amplification and suppression of thermoacoustic instabilities. Adding another layer of complexity, we then take into account nonlinearities in the flame response and the aeroacoustic coupling. The nonlinear dynamics of a symmetric model combustor are explored by exploiting the gradient structure of the averaged slow-flow dynamics. We obtain exact analytical expressions for the steady-state statistics, highlighting the connection between different emergent patterns and the resistive coupling between the cans. We leverage our analysis to explain the intermittent energy transfer between Bloch modes observed in real-world gas turbines.

Statistics of Weibull Record-Breaking Events

The statistics of record-breaking events plays an important role in the analysis of natural physical systems. It can provide an additional insight into the mechanisms and the occurrence of extreme events. In this work, the statistical aspects of the record-breaking events drawn from the Weibull distribution are considered and analyzed in detail. It is assumed that the underlying sequences of events are independent and identically distributed (i.i.d.). Several statistical measures of record-breaking events are analyzed. Exact analytical expressions are derived for the statistics of records. Particularly, the distributions of record magnitudes and the corresponding average magnitudes of records in case of Weibull distributed events are derived exactly for any specific record order and time step. In addition, a convolution operation is used to derive a recursive formula for the distribution of times of the occurrence of records. The analytical results are compared with the Monte Carlo simulations and their validity is confirmed. The numerical simulations also reveal that the finite-size effects strongly affect the statistics of records and need to be considered during the analysis of numerical experiments or empirical data.