Search research articles

ABOUT JoVE

Overview Leadership Blog JoVE Help Center

AUTHORS

Publishing Process Editorial Board Scope & Policies Peer Review FAQ Submit

LIBRARIANS

Testimonials Subscriptions Access Resources Library Advisory Board FAQ

RESEARCH

JoVE Journal Methods Collections JoVE Encyclopedia of Experiments Archive

EDUCATION

JoVE Core JoVE Business JoVE Science Education JoVE Lab Manual Faculty Resource Center Faculty Site

Terms & Conditions of Use

Related Concept Videos

Linear Approximation in Time Domain

Linear Approximation in Time Domain

Nonlinear systems often require sophisticated approaches for accurate modeling and analysis, with state-space representation being particularly effective. This method is especially useful for systems where variables and parameters vary with time or operating conditions, such as in a simple pendulum or a translational mechanical system with nonlinear springs.
For a simple pendulum with a mass evenly distributed along its length and the center of mass located at half the pendulum's length, the...

Partial Differential Equations

Partial Differential Equations

A stone dropped into a still pond generates waves that propagate outward in circular patterns, creating a dynamic surface whose elevation depends on both position and time. At any given location, the water level oscillates as the wave passes, while at any fixed moment, the surface exhibits smooth, curved structures extending across space. This dual dependence requires a mathematical description that accounts for variation in multiple variables simultaneously.At a fixed point on the water...

Reaction Mechanisms: The Steady-State Approximation

Reaction Mechanisms: The Steady-State Approximation

The steady-state approximation, also referred to as the quasi-steady-state approximation to differentiate it from a true steady state, is a widely used method for simplifying calculations in complex reaction mechanisms. This approach is particularly useful when dealing with multi-step reactions that involve reverse reactions or several steps, which can significantly increase mathematical complexity and make the reactions nearly unsolvable analytically.The steady-state approximation operates on...

Multivariable Chain Rule

Multivariable Chain Rule

When a variable z depends on two intermediate variables, x and y, and both x and y vary with respect to a third variable t, the dependence of z on t is indirect. Although t does not explicitly appear in the expression for z, any change in t produces corresponding changes in x and y, which in turn alter the value of z. The objective is to determine the total derivative of z with respect to t, denoted as dz/dt.Assuming that all functions involved are differentiable, the total change in z can be...

Change of Variables in Multiple Integrals

Change of Variables in Multiple Integrals

Multiple integrals are often used to evaluate areas, volumes, mass distributions, and other physical quantities over regions in two or three dimensions. In many problems, however, the original region may have complicated curved boundaries when expressed in Cartesian coordinates. These complex boundaries can make the limits of integration difficult to describe and the overall calculation cumbersome. To simplify the evaluation process, a change of variables is introduced that transforms the...

Approximate Integration

Approximate Integration

In many practical and theoretical contexts, the exact value of a definite integral may be inaccessible. This limitation typically arises when the antiderivative of a function is either unknown or cannot be expressed in a closed mathematical form. Alternatively, it can occur when a function is defined not by a formula but by a finite set of empirical data points, such as those collected during experiments. In these cases, approximate integration techniques provide a valuable solution.One of the...

You might also read

Related Articles

Articles linked to this work by shared authors, journal, and citation graph.

Sort by

Same author

Quadratic unconstrained binary optimization formulation for rectified-linear-unit-type functions.

Physical review. E·2019

Same author

Duality-based calculations for transition probabilities in stochastic chemical reactions.

Physical review. E·2017

Same author

Nonlinear Kalman filter based on duality relations between continuous and discrete-state stochastic processes.

Physical review. E, Statistical, nonlinear, and soft matter physics·2015

Same author

Stability of octreotide acetate decreases in a sodium bisulfate concentration-dependent manner: compatibility study with morphine and metoclopramide injections.

European journal of hospital pharmacy : science and practice·2015

Same author

One-parameter extension of the Doi-Peliti formalism and its relation with orthogonal polynomials.

Physical review. E, Statistical, nonlinear, and soft matter physics·2012

Same author

Counting statistics for genetic switches based on effective interaction approximation.

The Journal of chemical physics·2012

Search research articles

Related Experiment Video

Updated: Jul 3, 2026

Analyzing Melts and Fluids from Ab Initio Molecular Dynamics Simulations with the UMD Package

Analyzing Melts and Fluids from Ab Initio Molecular Dynamics Simulations with the UMD Package

Published on: September 17, 2021

Approximation scheme for master equations: Variational approach to multivariate case.

¹Institute for Solid State Physics, University of Tokyo, Kashiwanoha 5-1-5, Kashiwa-shi, Chiba 277-8581, Japan. ohkubo@issp.u-tokyo.ac.jp

The Journal of Chemical Physics

|August 7, 2008

Summary

This study introduces an improved approximation scheme for chemical master equations using second quantization and variational methods. The new approach accurately models small systems while reducing computational costs for complex reactions.

Related Experiment Videos

Last Updated: Jul 3, 2026

Analyzing Melts and Fluids from Ab Initio Molecular Dynamics Simulations with the UMD Package

Analyzing Melts and Fluids from Ab Initio Molecular Dynamics Simulations with the UMD Package

Published on: September 17, 2021

Area of Science:

Computational chemistry
Biophysical modeling
Chemical kinetics

Background:

Traditional rate equation approaches fail for small systems (e.g., cells) due to significant fluctuation effects.
Fokker-Planck equations, while accounting for fluctuations, incur high computational costs for complex chemical reactions.
System size expansion methods neglect the discrete nature of the original master equation.

Purpose of the Study:

To develop an efficient approximation scheme for the chemical master equation.
To address limitations of existing methods in modeling small, fluctuating chemical systems.
To propose a new variational function selection scheme for improved accuracy and efficiency.

Main Methods:

Utilizing a second quantization description of the chemical master equation.
Applying a variational method for dimensionality reduction.
Introducing a novel scheme for selecting variational functions applicable to multivariate systems.

Main Results:

The proposed scheme significantly reduces the dimensionality of the master equation while preserving discrete characteristics.
The new variational function selection method yields superior numerical results compared to previous approaches.
The computational cost of the new scheme increases only slightly, offering a favorable trade-off.

Conclusions:

The developed approximation scheme provides an accurate and computationally efficient method for studying chemical master equations.
The new variational function selection strategy enhances the applicability and performance of the second quantization approach.
This method offers a promising alternative for modeling stochastic chemical kinetics in small systems.