Quantum thermodynamic uncertainty relation for continuous measurement

We use quantum estimation theory to derive a thermodynamic uncertainty relation in Markovian open quantum systems, which bounds the fluctuation of continuous measurements. The derived quantum thermodynamic uncertainty relation holds for arbitrary continuous measurements satisfying a scaling condition. We derive two relations; the first relation bounds the fluctuation by the dynamical activity and the second one does so by the entropy production. We apply our bounds to a two-level atom driven by a laser field and a three-level quantum thermal machine with jump and diffusion measurements. Our result shows that there exists a universal bound upon the fluctuations, regardless of continuous measurements.Comment: 6 pages, 3 figures; 10 pages of supplemental material with 2 figure

Multi-dimensional biochemical information processing of dynamical patterns

Cells receive signaling molecules by receptors and relay information via sensory networks so that they can respond properly depending on the type of signal. Recent studies have shown that cells can extract multi-dimensional information from dynamical concentration patterns of signaling molecules. We herein study how biochemical systems can process multi-dimensional information embedded in dynamical patterns. We model the decoding networks by linear response functions, and optimize the functions with the calculus of variations to maximize the mutual information between patterns and output. We find that, when the noise intensity is lower, decoders with different linear response functions, i.e., distinct decoders, can extract much information. However, when the noise intensity is higher, distinct decoders do not provide the maximum amount of information. This indicates that, when transmitting information by dynamical patterns, embedding information in multiple patterns is not optimal when the noise intensity is very large. Furthermore, we explore the biochemical implementations of these decoders using control theory and demonstrate that these decoders can be implemented biochemically through the modification of cascade-type networks, which are prevalent in actual signaling pathways.Comment: 12 pages, 7 figure

Optimal temporal patterns for dynamical cellular signaling

Cells use temporal dynamical patterns to transmit information via signaling pathways. As optimality with respect to the environment plays a fundamental role in biological systems, organisms have evolved optimal ways to transmit information. Here, we use optimal control theory to obtain the dynamical signal patterns for the optimal transmission of information, in terms of efficiency (low energy) and reliability (low uncertainty). Adopting an activation-deactivation decoding network, we reproduce several dynamical patterns found in actual signals, such as steep, gradual, and overshooting dynamics. Notably, when minimizing the energy of the input signal, the optimal signals exhibit overshooting, which is a biphasic pattern with transient and steady phases; this pattern is prevalent in actual dynamical patterns. We also identify conditions in which these three patterns (steep, gradual, and overshooting) confer advantages. Our study shows that cellular signal transduction is governed by the principle of minimizing free energy dissipation and uncertainty; these constraints serve as selective pressures when designing dynamical signaling patterns.Comment: 12 pages, 5 figures; 5 pages of supplemental material with 1 figur

Variational superposed Gaussian approximation for time-dependent solutions of Langevin equations

We propose a variational superposed Gaussian approximation (VSGA) for dynamical solutions of Langevin equations subject to applied signals, determining time-dependent parameters of superposed Gaussian distributions by the variational principle. We apply the proposed VSGA to systems driven by a chaotic signal, where the conventional Fourier method cannot be adopted, and calculate the time evolution of probability density functions (PDFs) and moments. Both white and colored Gaussian noises terms are included to describe fluctuations. Our calculations show that time-dependent PDFs obtained by VSGA agree excellently with those obtained by Monte Carlo simulations. The correlation between the chaotic input signal and the mean response are also calculated as a function of the noise intensity, which confirms the occurrence of aperiodic stochastic resonance with both white and colored noises.Comment: 22 pages, 12 figure

Thermodynamics of collective enhancement of precision

The circadian oscillator exhibits remarkably high temporal precision, despite its exposure to several fluctuations. The central mechanism that protects the oscillator from fluctuations is a collective enhancement of precision, where a population of coupled oscillators displays higher temporal precision than that achieved without coupling. Since coupling is essentially information exchange between oscillators, we herein investigate the relation between the temporal precision and the information flow between oscillators in the linearized Kuramoto model by using stochastic thermodynamics. For general coupling, we find that the temporal precision is bounded from below by the information flow. We generalize the model to incorporate a time-delayed coupling and demonstrate that the same relation also holds for the time-delayed case. Furthermore, the temporal precision is demonstrated to be improved in the presence of the time delay, and we show that the increased information flow is responsible for the time-delay-induced precision improvement.Comment: 9 pages, 4 figure

Noise-intensity fluctuation in Langevin model and its higher-order Fokker-Planck equation

In this paper, we investigate a Langevin model subjected to stochastic intensity noise (SIN), which incorporates temporal fluctuations in noise-intensity. We derive a higher-order Fokker-Planck equation (HFPE) of the system, taking into account the effect of SIN by the adiabatic elimination technique. Stationary distributions of the HFPE are calculated by using the perturbation expansion. We investigate the effect of SIN in three cases: (a) parabolic and quartic bistable potentials with additive noise, (b) a quartic potential with multiplicative noise, and (c) a stochastic gene expression model. We find that the existence of noise intensity fluctuations induces an intriguing phenomenon of a bimodal-to-trimodal transition in probability distributions. These results are validated with Monte Carlo simulations.Comment: 23 pages, 7 figures, 1 tabl

Approximate Vanishing Ideal via Data Knotting

The vanishing ideal is a set of polynomials that takes zero value on the given data points. Originally proposed in computer algebra, the vanishing ideal has been recently exploited for extracting the nonlinear structures of data in many applications. To avoid overfitting to noisy data, the polynomials are often designed to approximately rather than exactly equal zero on the designated data. Although such approximations empirically demonstrate high performance, the sound algebraic structure of the vanishing ideal is lost. The present paper proposes a vanishing ideal that is tolerant to noisy data and also pursued to have a better algebraic structure. As a new problem, we simultaneously find a set of polynomials and data points for which the polynomials approximately vanish on the input data points, and almost exactly vanish on the discovered data points. In experimental classification tests, our method discovered much fewer and lower-degree polynomials than an existing state-of-the-art method. Consequently, our method accelerated the runtime of the classification tasks without degrading the classification accuracy.Comment: 11 pages; AAAI'1

Optimal implementations for reliable circadian clocks

Circadian rhythms are acquired through evolution to increase the chances for survival through synchronizing with the daylight cycle. Reliable synchronization is realized through two trade-off properties: regularity to keep time precisely, and entrainability to synchronize the internal time with daylight. We found by using a phase model with multiple inputs that achieving the maximal limit of regularity and entrainability entails many inherent features of the circadian mechanism. At the molecular level, we demonstrate the role sharing of two light inputs, phase advance and delay, as is well observed in mammals. At the behavioral level, the optimal phase-response curve inevitably contains a dead zone, a time during which light pulses neither advance nor delay the clock. We reproduce the results of phase-controlling experiments entrained by two types of periodic light pulses. Our results indicate that circadian clocks are designed optimally for reliable clockwork through evolution.Comment: 5 pages, 4 figures; 6 pages of supplemental material with 2 figure

Escape process and stochastic resonance under noise-intensity fluctuation

We study the effects of noise-intensity fluctuations on the stationary and dynamical properties of an overdamped Langevin model with a bistable potential and external periodical driving force. We calculated the stationary distributions, mean-first passage time (MFPT) and the spectral amplification factor using a complete set expansion (CSE) technique. We found resonant activation (RA) and stochastic resonance (SR) phenomena in the system under investigation. Moreover, the strength of RA and SR phenomena exhibit non-monotonic behavior and their trade-off relation as a function of the squared variation coefficient of the noise-intensity process. The reliability of CSE is verified with Monte Carlo simulations.Comment: 23 pages, 7 figure

Augmented Variational Superposed Gaussian Approximation for Langevin Equations with Rational Polynomial Functions

Reliable methods for obtaining time-dependent solutions of Langevin equations are in high demand in the field of non-equilibrium theory. In this paper, we present a new method based on variational superposed Gaussian approximation (VSGA) and Pad\'e approximant. The VSGA obtains time-dependent probability density functions as a superposition of multiple Gaussian distributions. However, a limitation of the VSGA is that the expectation of the drift term with respect to the Gaussian distribution should be calculated analytically, which is typically satisfied when the drift term is a polynomial function. When this condition is not met, the VSGA must rely on the numerical integration of the expectation at each step, resulting in huge computational cost. We propose an augmented VSGA (A-VSGA) method that effectively overcomes the limitation of the VSGA by approximating non-linear functions with the Pad\'e approximant. We apply the A-VSGA to two systems driven by chaotic input signals, a stochastic genetic regulatory system and a soft bistable system, whose drift terms are a rational polynomial function and a hyperbolic tangent function, respectively. The numerical calculations show that the proposed method can provide accurate results with less temporal cost than that required for Monte Carlo simulation.Comment: 10 pages, 9 figure