Rare events play a crucial role in understanding complex systems. Characterizing and analyzing them in scale-invariant situations is challenging due to strong correlations. In this work, we focus on characterizing the tails of probability distribution functions (PDFs) for these systems. Using a variety of methods, perturbation theory, functional renormalization group, hierarchical models, large $n$ limit, and Monte Carlo simulations, we investigate universal rare events of critical $O(n)$ systems. Additionally, we explore the crossover from universal to nonuniversal behavior in PDF tails, extending Cram'er's series to strongly correlated variables. Our findings highlight the universal and nonuniversal aspects of rare event statistics and challenge existing assumptions about power-law corrections to the leading stretched exponential decay in these tails.
稀有事件在理解复杂系统中起着至关重要的作用。由于存在强相关性，在规模不变的情况下描述和分析稀有事件极具挑战性。在这项工作中，我们重点研究这些系统的概率分布函数（PDF）尾部的特征。我们使用多种方法，包括扰动理论、泛函重正化群、层次模型、大 $n$ 极限和蒙特卡罗模拟，研究临界 $O(n)$ 系统的普遍罕见事件。此外，我们还探索了 PDF 尾部从普遍到非普遍行为的交叉，并将 Cram'er's series 扩展到强相关变量。我们的发现突出了罕见事件统计的普遍性和非普遍性，并挑战了现有的关于在这些尾部对前导拉伸指数衰减进行幂律修正的假设。
{"title":"Universal and non-universal large deviations in critical systems","authors":"Ivan Balo, Bertrand Delamotte, Adam Rançon","doi":"arxiv-2409.01250","DOIUrl":"https://doi.org/arxiv-2409.01250","url":null,"abstract":"Rare events play a crucial role in understanding complex systems.\u0000Characterizing and analyzing them in scale-invariant situations is challenging\u0000due to strong correlations. In this work, we focus on characterizing the tails\u0000of probability distribution functions (PDFs) for these systems. Using a variety\u0000of methods, perturbation theory, functional renormalization group, hierarchical\u0000models, large $n$ limit, and Monte Carlo simulations, we investigate universal\u0000rare events of critical $O(n)$ systems. Additionally, we explore the crossover\u0000from universal to nonuniversal behavior in PDF tails, extending Cram'er's\u0000series to strongly correlated variables. Our findings highlight the universal\u0000and nonuniversal aspects of rare event statistics and challenge existing\u0000assumptions about power-law corrections to the leading stretched exponential\u0000decay in these tails.","PeriodicalId":501520,"journal":{"name":"arXiv - PHYS - Statistical Mechanics","volume":"57 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142196241","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The critical phase boundary of a system, in general, can depend on one or more parameters. We show that by calculating the Gini index ($g$) of any suitably defined response function of a system, the critical phase boundary can always be reduced to that of a single parameter, starting from $g=0$ and terminating at $g=g_f$, where $g_f$ is a universal number for a given universality class. We demonstrate the construction with analytical and numerical calculations of mean field transverse field Ising model and site diluted Ising model on the Bethe lattice, respectively. Both models have two parameter phase boundaries -- transverse field and Temperature for the first case and site dilution and temperature in the second case. Both can be reduced to single parameter transition points in terms of the Gini index. The method is generally applicable for any multi-parameter critical transition.
{"title":"Universal critical phase diagram using Gini index","authors":"Soumyaditya Das, Soumyajyoti Biswas","doi":"arxiv-2409.01453","DOIUrl":"https://doi.org/arxiv-2409.01453","url":null,"abstract":"The critical phase boundary of a system, in general, can depend on one or\u0000more parameters. We show that by calculating the Gini index ($g$) of any\u0000suitably defined response function of a system, the critical phase boundary can\u0000always be reduced to that of a single parameter, starting from $g=0$ and\u0000terminating at $g=g_f$, where $g_f$ is a universal number for a given\u0000universality class. We demonstrate the construction with analytical and\u0000numerical calculations of mean field transverse field Ising model and site\u0000diluted Ising model on the Bethe lattice, respectively. Both models have two\u0000parameter phase boundaries -- transverse field and Temperature for the first\u0000case and site dilution and temperature in the second case. Both can be reduced\u0000to single parameter transition points in terms of the Gini index. The method is\u0000generally applicable for any multi-parameter critical transition.","PeriodicalId":501520,"journal":{"name":"arXiv - PHYS - Statistical Mechanics","volume":"4 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142196239","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Recently, the full version of the Eigenstate Thermalization Hypothesis (ETH) has been systematized using Free Probability. In this paper, we present a detailed discussion of the Free Cumulants approach to many-body dynamics within the microcanonical ensemble. Differences between the later and canonical averages are known to manifest in the time-dependent fluctuations of extensive operators. Thus, the microcanonical ensemble is essential to extend the application of Free Probability to the broad class of extensive observables. We numerically demonstrate the validity of our approach in a non-integrable spin chain Hamiltonian for extensive observables at finite energy density. Our results confirm the full ETH properties, specifically the suppression of crossing contributions and the factorization of non-crossing ones, thus demonstrating that the microcanonical free cumulants encode ETH smooth correlations for both local and extensive observables.
最近，利用自由概率对完整版的特征态热化假说（ETH）进行了系统化。在本文中，我们详细讨论了微规范集合中的多体动力学自由累计数方法。众所周知，后期平均与经典平均之间的差异表现在广泛运算器随时间变化的波动上。因此，微规范集合对于将自由概率的应用扩展到广泛的可观测变量类别至关重要。在有限能量密度下，我们用数值证明了我们的方法在广义可观测量的不可积分自旋链哈密顿中的有效性。我们的结果证实了完整的 ETH 特性，特别是抑制了交叉贡献和非交叉贡献的因子化，从而证明微观经典自由积累编码了局部和广泛观测值的 ETH 平滑相关性。
{"title":"Microcanonical Free Cumulants in lattice systems","authors":"Felix Fritzsch, Tomaž Prosen, Silvia Pappalardi","doi":"arxiv-2409.01404","DOIUrl":"https://doi.org/arxiv-2409.01404","url":null,"abstract":"Recently, the full version of the Eigenstate Thermalization Hypothesis (ETH)\u0000has been systematized using Free Probability. In this paper, we present a\u0000detailed discussion of the Free Cumulants approach to many-body dynamics within\u0000the microcanonical ensemble. Differences between the later and canonical\u0000averages are known to manifest in the time-dependent fluctuations of extensive\u0000operators. Thus, the microcanonical ensemble is essential to extend the\u0000application of Free Probability to the broad class of extensive observables. We\u0000numerically demonstrate the validity of our approach in a non-integrable spin\u0000chain Hamiltonian for extensive observables at finite energy density. Our\u0000results confirm the full ETH properties, specifically the suppression of\u0000crossing contributions and the factorization of non-crossing ones, thus\u0000demonstrating that the microcanonical free cumulants encode ETH smooth\u0000correlations for both local and extensive observables.","PeriodicalId":501520,"journal":{"name":"arXiv - PHYS - Statistical Mechanics","volume":"24 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142196242","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The physical significance of the stochastic processes associated to the generalized Gibbs ensembles is scrutinized here with special attention to the thermodynamic fluctuations of small systems. The contact with the environment produces an interaction entropy, which controls the distribution of fluctuations and allows writing the generalized Gibbs ensembles for macrostates in potential form. This naturally yields exact nonlinear thermodynamic Langevin equations (TLEs) for such variables, with drift expressed in terms of entropic forces. The analysis of the canonical ensemble for an ideal monoatomic gas and the related TLEs show that introducing currents leads to nonequilibrium heat transfer conditions with interesting bounds on entropy production but with no obvious thermodynamic limit. For a colloidal particle under constant force, the TLEs for macroscopic variables are different from those for the microscopic position, typically used in the so-called stochastic thermodynamics; while TLEs are consistent with the fundamental equation obtained from the Hamiltonian, stochastic thermodynamics requires isothermal conditions and entropy proportional to position.
本文仔细研究了与广义吉布斯集合相关的随机过程的物理意义，并特别关注小系统的热力学波动。与环境的接触产生了相互作用熵，它控制着波动的分布，并允许以势能形式写出宏观状态的广义吉布斯集合。这自然会产生此类变量的精确非线性热力学兰格方程（TLEs），其漂移用熵力表示。对理想单原子气体的典型集合和相关 TLE 的分析表明，引入电流会导致非平衡传热条件，对熵的产生有有趣的限制，但没有明显的热力学极限。对于恒力作用下的胶体粒子，宏观变量的 TLE 与所谓随机热力学通常使用的微观位置变量的 TLE 不同；TLE 与从哈密顿方程得到的基本方程一致，而随机热力学需要等温条件和与位置成比例的熵。
{"title":"Thermodynamic Langevin Equations","authors":"Amilcare Porporato, Salvatore Calabrese, Lamberto Rondoni","doi":"arxiv-2409.00811","DOIUrl":"https://doi.org/arxiv-2409.00811","url":null,"abstract":"The physical significance of the stochastic processes associated to the\u0000generalized Gibbs ensembles is scrutinized here with special attention to the\u0000thermodynamic fluctuations of small systems. The contact with the environment\u0000produces an interaction entropy, which controls the distribution of\u0000fluctuations and allows writing the generalized Gibbs ensembles for macrostates\u0000in potential form. This naturally yields exact nonlinear thermodynamic Langevin\u0000equations (TLEs) for such variables, with drift expressed in terms of entropic\u0000forces. The analysis of the canonical ensemble for an ideal monoatomic gas and\u0000the related TLEs show that introducing currents leads to nonequilibrium heat\u0000transfer conditions with interesting bounds on entropy production but with no\u0000obvious thermodynamic limit. For a colloidal particle under constant force, the\u0000TLEs for macroscopic variables are different from those for the microscopic\u0000position, typically used in the so-called stochastic thermodynamics; while TLEs\u0000are consistent with the fundamental equation obtained from the Hamiltonian,\u0000stochastic thermodynamics requires isothermal conditions and entropy\u0000proportional to position.","PeriodicalId":501520,"journal":{"name":"arXiv - PHYS - Statistical Mechanics","volume":"66 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142196260","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study percolation problems of overlapping objects where the underlying geometry is such that in D-dimensions, a subset of the directions has a lattice structure, while the remaining directions have a continuum structure. The resulting semicontinuum problem describes the percolation of overlapping shapes in parallel layers or lanes with positional constraints for the placement of the objects along the discrete directions. Several semicontinuum percolation systems are analyzed like hypercuboids with a particular focus on 2D and 3D cases, disks, and parallelograms. Adapting the excluded volume arguments to the semicontinuum setting, we show that for the semicontinuum problem of hypercuboids, for fixed side-lengths of the hypercuboids along the directions in which a lattice structure is maintained, the percolation threshold is always independent of the side-lengths along the continuum directions. The result holds even when there is a distribution for the side-lengths along the continuum directions. Trends in the variation of the thresholds, as we vary the linear measure of the shapes along the continuum directions, are obtained for other semicontinuum models like disks and parallelograms in 2D. The results are compared with those of corresponding continuum and lattice models. For the 2D and 3D models considered, using Monte Carlo simulations, we verify the excluded volume predictions for the trends and numerical values of the percolation thresholds. Very good agreement is seen between the predicted numerical values and the simulation results. The semicontinuum setting also allows us to establish a connection between the percolation problem of overlapping shapes in 2D continuum and triangular lattice. We also verify that the isotropy of the threshold for anisotropic shapes and standard percolation universality class is maintained in the semicontinuum setting.
我们研究的是重叠物体的渗滤问题，在这些物体的底层几何中，在 D 维中，一个方向子集具有晶格结构，而其余方向具有连续结构。由此产生的半连续问题描述了重叠图形在平行层或平行线上的渗滤，并对物体沿离散方向的位置进行了限制。我们分析了几种半真空渗滤系统，如超立方体，尤其侧重于二维和三维案例、圆盘和平行四边形。通过将排除体积论证应用于半真空环境，我们证明了对于超立方体的半真空问题，对于超立方体沿保持晶格结构的方向的固定边长，渗流阈值总是与沿连续方向的边长无关。即使沿连续方向的边长存在分布，结果也是如此。当我们沿连续方向改变形状的线性度量时，阈值的变化趋势也会随之改变，这适用于其他半连续模型，如二维中的圆盘和平行四边形。这些结果与相应的连续体和晶格模型的结果进行了比较。对于所考虑的二维和三维模型，我们利用蒙特卡罗模拟验证了排除体积对渗流阈值趋势和数值的预测。结果表明，预测数值与模拟结果非常吻合。半连续性设置还使我们能够在二维连续性和三角形晶格中重叠形状的渗滤问题之间建立联系。我们还验证了各向异性形状阈值的各向同性和标准渗滤普遍性类在半真空环境中得以保持。
{"title":"Percolation in semicontinuum geometries","authors":"Jasna C. K, V. Krishnadev, V. Sasidevan","doi":"arxiv-2409.00699","DOIUrl":"https://doi.org/arxiv-2409.00699","url":null,"abstract":"We study percolation problems of overlapping objects where the underlying\u0000geometry is such that in D-dimensions, a subset of the directions has a lattice\u0000structure, while the remaining directions have a continuum structure. The\u0000resulting semicontinuum problem describes the percolation of overlapping shapes\u0000in parallel layers or lanes with positional constraints for the placement of\u0000the objects along the discrete directions. Several semicontinuum percolation\u0000systems are analyzed like hypercuboids with a particular focus on 2D and 3D\u0000cases, disks, and parallelograms. Adapting the excluded volume arguments to the\u0000semicontinuum setting, we show that for the semicontinuum problem of\u0000hypercuboids, for fixed side-lengths of the hypercuboids along the directions\u0000in which a lattice structure is maintained, the percolation threshold is always\u0000independent of the side-lengths along the continuum directions. The result\u0000holds even when there is a distribution for the side-lengths along the\u0000continuum directions. Trends in the variation of the thresholds, as we vary the\u0000linear measure of the shapes along the continuum directions, are obtained for\u0000other semicontinuum models like disks and parallelograms in 2D. The results are\u0000compared with those of corresponding continuum and lattice models. For the 2D\u0000and 3D models considered, using Monte Carlo simulations, we verify the excluded\u0000volume predictions for the trends and numerical values of the percolation\u0000thresholds. Very good agreement is seen between the predicted numerical values\u0000and the simulation results. The semicontinuum setting also allows us to\u0000establish a connection between the percolation problem of overlapping shapes in\u00002D continuum and triangular lattice. We also verify that the isotropy of the\u0000threshold for anisotropic shapes and standard percolation universality class is\u0000maintained in the semicontinuum setting.","PeriodicalId":501520,"journal":{"name":"arXiv - PHYS - Statistical Mechanics","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142196261","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the problem of a run and tumble particle in a harmonic trap, with a finite run and tumble time, by a direct integration of the equation of motion. An exact 1D steady state distribution, diagram laws and a programmable Volterra difference equation are derived to calculate any order of moments in any other dimension, both for steady state as well as the Laplace transform in time for the intermediate states. We also use the moments to infer the distribution by considering a Gaussian quadrature for the corresponding measure, and from the scaling law of high order moments.
{"title":"Exact moments for a run and tumble particle in a harmonic trap with a finite tumble time","authors":"Aoran Sun, Fangfu Ye, Rudolf Podgornik","doi":"arxiv-2409.00578","DOIUrl":"https://doi.org/arxiv-2409.00578","url":null,"abstract":"We study the problem of a run and tumble particle in a harmonic trap, with a\u0000finite run and tumble time, by a direct integration of the equation of motion.\u0000An exact 1D steady state distribution, diagram laws and a programmable Volterra\u0000difference equation are derived to calculate any order of moments in any other\u0000dimension, both for steady state as well as the Laplace transform in time for\u0000the intermediate states. We also use the moments to infer the distribution by\u0000considering a Gaussian quadrature for the corresponding measure, and from the\u0000scaling law of high order moments.","PeriodicalId":501520,"journal":{"name":"arXiv - PHYS - Statistical Mechanics","volume":"25 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142196265","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The behavior of lattice models in which time reversibility is enforced at the level of trajectories (microscopic reversibility) is studied analytically. Conditions for ergodicity breaking are explored, and a few examples of systems characterized by an additional conserved quantity besides energy are presented. All the systems are characterized by ergodicity restoration when put in contact with a thermal bath, except for specific choices of the interactions between the atoms in the system and the bath. The study shows that the additional conserved quantities return to play a role in non-equilibrium conditions, with behaviors similar to those of some mesoscale systems, in which the transition rates satisfy detailed balance but not microscopic reversibility.
{"title":"Models of heat transport with microscopic reversibility","authors":"Piero Olla","doi":"arxiv-2409.00430","DOIUrl":"https://doi.org/arxiv-2409.00430","url":null,"abstract":"The behavior of lattice models in which time reversibility is enforced at the\u0000level of trajectories (microscopic reversibility) is studied analytically.\u0000Conditions for ergodicity breaking are explored, and a few examples of systems\u0000characterized by an additional conserved quantity besides energy are presented.\u0000All the systems are characterized by ergodicity restoration when put in contact\u0000with a thermal bath, except for specific choices of the interactions between\u0000the atoms in the system and the bath. The study shows that the additional\u0000conserved quantities return to play a role in non-equilibrium conditions, with\u0000behaviors similar to those of some mesoscale systems, in which the transition\u0000rates satisfy detailed balance but not microscopic reversibility.","PeriodicalId":501520,"journal":{"name":"arXiv - PHYS - Statistical Mechanics","volume":"181 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142196262","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For a Markov and stationary stochastic process described by the well-known classical master equation, we introduce complex transition rates instead of real transition rates to study the pre-thermal oscillatory behaviour in complex probabilities. Further, for purely imaginary transition rates we obtain persistent infinitely long lived oscillations in complex probability whose nature depends on the dimensionality of the state space. We also take a peek into cases where we perturb the relaxation matrix for a dichotomous process with an oscillatory drive where the relative sign of the angular frequency of the drive decides whether there will be dissipation in the complex probability or not.
{"title":"Oscillatory and dissipative dynamics of complex probability in non-equilibrium stochastic processes","authors":"Anwesha Chattopadhyay","doi":"arxiv-2409.00361","DOIUrl":"https://doi.org/arxiv-2409.00361","url":null,"abstract":"For a Markov and stationary stochastic process described by the well-known\u0000classical master equation, we introduce complex transition rates instead of\u0000real transition rates to study the pre-thermal oscillatory behaviour in complex\u0000probabilities. Further, for purely imaginary transition rates we obtain\u0000persistent infinitely long lived oscillations in complex probability whose\u0000nature depends on the dimensionality of the state space. We also take a peek\u0000into cases where we perturb the relaxation matrix for a dichotomous process\u0000with an oscillatory drive where the relative sign of the angular frequency of\u0000the drive decides whether there will be dissipation in the complex probability\u0000or not.","PeriodicalId":501520,"journal":{"name":"arXiv - PHYS - Statistical Mechanics","volume":"70 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142196263","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Petrus H. R. dos Anjos, Fernando A. Oliveira, David L. Azevedo
We propose two new kinds of infinite resistor networks based on the Fibonacci sequence: a serial association of resistor sets connected in parallel (type 1) or a parallel association of resistor sets connected in series (type 2). We show that the sequence of the network's equivalent resistance converges uniformly in the parameter $alpha=frac{r_2}{r_1} in [0,+infty)$, where $r_1$ and $r_2$ are the first and second resistors in the network. We also show that these networks exhibit self-similarity and scale invariance, which mimics a self-similar fractal. We also provide some generalizations, including resistor networks based on high-order Fibonacci sequences and other recursive combinatorial sequences.
{"title":"Fractality in resistive circuits: The Fibonacci resistor networks","authors":"Petrus H. R. dos Anjos, Fernando A. Oliveira, David L. Azevedo","doi":"arxiv-2409.00229","DOIUrl":"https://doi.org/arxiv-2409.00229","url":null,"abstract":"We propose two new kinds of infinite resistor networks based on the Fibonacci\u0000sequence: a serial association of resistor sets connected in parallel (type 1)\u0000or a parallel association of resistor sets connected in series (type 2). We\u0000show that the sequence of the network's equivalent resistance converges\u0000uniformly in the parameter $alpha=frac{r_2}{r_1} in [0,+infty)$, where\u0000$r_1$ and $r_2$ are the first and second resistors in the network. We also show\u0000that these networks exhibit self-similarity and scale invariance, which mimics\u0000a self-similar fractal. We also provide some generalizations, including\u0000resistor networks based on high-order Fibonacci sequences and other recursive\u0000combinatorial sequences.","PeriodicalId":501520,"journal":{"name":"arXiv - PHYS - Statistical Mechanics","volume":"42 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142196264","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Frank Barrows, Jonathan Lin, Francesco Caravelli, Dante R. Chialvo
Hardware-based neuromorphic computing remains an elusive goal with the potential to profoundly impact future technologies and deepen our understanding of emergent intelligence. The learning-from-mistakes algorithm is one of the few training algorithms inspired by the brain's simple learning rules, utilizing inhibition and pruning to demonstrate self-organized learning. Here we implement this algorithm in purely neuromorphic memristive hardware through a co-design process. This implementation requires evaluating hardware trade-offs and constraints. It has been shown that learning-from-mistakes successfully trains small networks to function as binary classifiers and perceptrons. However, without tailoring the hardware to the algorithm, performance decreases exponentially as the network size increases. When implementing neuromorphic algorithms on neuromorphic hardware, we investigate the trade-offs between depth, controllability, and capacity, the latter being the number of learnable patterns. We emphasize the significance of topology and the use of governing equations, demonstrating theoretical tools to aid in the co-design of neuromorphic hardware and algorithms. We provide quantitative techniques to evaluate the computational capacity of a neuromorphic device based on the measurements performed and the underlying circuit structure. This approach shows that breaking the symmetry of a neural network can increase both the controllability and average network capacity. By pruning the circuit, neuromorphic algorithms in all-memristive device circuits leverage stochastic resources to drive local contrast in network weights. Our combined experimental and simulation efforts explore the parameters that make a network suited for displaying emergent intelligence from simple rules.
{"title":"Uncontrolled learning: co-design of neuromorphic hardware topology for neuromorphic algorithms","authors":"Frank Barrows, Jonathan Lin, Francesco Caravelli, Dante R. Chialvo","doi":"arxiv-2408.05183","DOIUrl":"https://doi.org/arxiv-2408.05183","url":null,"abstract":"Hardware-based neuromorphic computing remains an elusive goal with the\u0000potential to profoundly impact future technologies and deepen our understanding\u0000of emergent intelligence. The learning-from-mistakes algorithm is one of the\u0000few training algorithms inspired by the brain's simple learning rules,\u0000utilizing inhibition and pruning to demonstrate self-organized learning. Here\u0000we implement this algorithm in purely neuromorphic memristive hardware through\u0000a co-design process. This implementation requires evaluating hardware\u0000trade-offs and constraints. It has been shown that learning-from-mistakes\u0000successfully trains small networks to function as binary classifiers and\u0000perceptrons. However, without tailoring the hardware to the algorithm,\u0000performance decreases exponentially as the network size increases. When\u0000implementing neuromorphic algorithms on neuromorphic hardware, we investigate\u0000the trade-offs between depth, controllability, and capacity, the latter being\u0000the number of learnable patterns. We emphasize the significance of topology and\u0000the use of governing equations, demonstrating theoretical tools to aid in the\u0000co-design of neuromorphic hardware and algorithms. We provide quantitative\u0000techniques to evaluate the computational capacity of a neuromorphic device\u0000based on the measurements performed and the underlying circuit structure. This\u0000approach shows that breaking the symmetry of a neural network can increase both\u0000the controllability and average network capacity. By pruning the circuit,\u0000neuromorphic algorithms in all-memristive device circuits leverage stochastic\u0000resources to drive local contrast in network weights. Our combined experimental\u0000and simulation efforts explore the parameters that make a network suited for\u0000displaying emergent intelligence from simple rules.","PeriodicalId":501520,"journal":{"name":"arXiv - PHYS - Statistical Mechanics","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141943341","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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