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Related Concept Videos

Entropy Change in Reversible Processes01:10

Entropy Change in Reversible Processes

In the Carnot engine, which achieves the maximum efficiency between two reservoirs of fixed temperatures, the total change in entropy is zero. The observation can be generalized by considering any reversible cyclic process consisting of many Carnot cycles. Thus, it can be stated that the total entropy change of any ideal reversible cycle is zero.
The statement can be further generalized to prove that entropy is a state function. Take a cyclic process between any two points on a p-V diagram.
Reversible and Irreversible Processes01:14

Reversible and Irreversible Processes

The thermodynamic processes can be classified into reversible and irreversible processes. The processes that can be restored to their initial state are called reversible processes. It is only possible if the process is in quasi-static equilibrium, i.e., it takes place in infinitesimally small steps, and the system remains at equilibrium However, these are ideal processes and do not occur naturally. An ideal system undergoing a reversible process is always in thermodynamic equilibrium within...
The Entropy as a State Function01:14

The Entropy as a State Function

Consider an arbitrary process that moves between two specific states (A and B) in a cyclic manner. This process is reversible and broken down into smaller parts that each follow a Carnot cycle. A Carnot cycle has two isothermal (constant temperature) processes. During these processes, the ratio of the amount of heat transferred to their respective temperature remains constant. The other two processes in the Carnot cycle are also reversible but adiabatic, which means they occur without any heat...
Entropy Changes Accompanying Specific Processes01:21

Entropy Changes Accompanying Specific Processes

Entropy, a measure of disorder in a system, changes during phase transitions like freezing or boiling. At the transition temperature Ttrs, where two phases are in equilibrium, the phase transition is a reversible process. The entropy change can be calculated from a substance's enthalpy of transition using the equation ΔStrs = ΔtrsH /Ttrs.When a perfect gas expands isothermally from one volume to another, entropy increases logarithmically with volume. Conversely, isothermal compression results...
BIBO stability of continuous and discrete -time systems01:24

BIBO stability of continuous and discrete -time systems

System stability is a fundamental concept in signal processing, often assessed using convolution. For a system to be considered bounded-input bounded-output (BIBO) stable, any bounded input signal must produce a bounded output signal. A bounded input signal is one where the modulus does not exceed a certain constant at any point in time.
To determine the BIBO stability, the convolution integral is utilized when a bounded continuous-time input is applied to a Linear Time-Invariant (LTI) system.
Cyclic Processes And Isolated Systems01:19

Cyclic Processes And Isolated Systems

A thermodynamic system with zero heat exchange and work is an isolated system. For these systems, the internal energy remains constant.
In the case of a non-isolated system, the change in the internal energy is zero only if the process is cyclic. A thermodynamic process is considered cyclic if the system undergoes a series of changes and returns to its initial state. 
Consider a cyclic process that returns to its initial state, undergoing a four-step process. The heat transfer along each path...

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Related Experiment Video

Updated: May 8, 2026

An In Vitro System to Study Tumor Dormancy and the Switch to Metastatic Growth
09:14

An In Vitro System to Study Tumor Dormancy and the Switch to Metastatic Growth

Published on: August 11, 2011

Metastable behavior in Markov processes with internal states.

Jay Newby1, Jon Chapman

  • 1Mathematical Institute, University of Oxford, 24-29 St Giles', Oxford, OX1 3LB, UK, newby@maths.ox.ac.uk.

Journal of Mathematical Biology
|September 3, 2013
PubMed
Summary

We developed a new perturbation framework to analyze metastable behavior in stochastic systems. This method accurately approximates switching times in bistable systems, including a gene expression model.

Related Experiment Videos

Last Updated: May 8, 2026

An In Vitro System to Study Tumor Dormancy and the Switch to Metastatic Growth
09:14

An In Vitro System to Study Tumor Dormancy and the Switch to Metastatic Growth

Published on: August 11, 2011

Area of Science:

  • * Mathematical modeling and theoretical physics.
  • * Computational biology and systems biology.

Background:

  • * Stochastic processes often exhibit metastable behavior, characterized by long-lived states separated by energy barriers.
  • * Analyzing this behavior is crucial for understanding complex systems like gene regulatory networks.
  • * Existing methods like diffusion approximations may not accurately capture long-time metastable dynamics.

Purpose of the Study:

  • * To develop a novel perturbation framework for analyzing metastable behavior in stochastic processes.
  • * To derive general analytical approximations for stationary probability density and mean switching times.
  • * To validate the framework using a well-established bistable gene expression model.

Main Methods:

  • * Development of a perturbation framework applicable to stochastic processes with internal and external states.
  • * Analysis under weak noise conditions and consideration of bistable deterministic limits.
  • * Derivation of analytical approximations for stationary probability density and mean switching time, including the pre-exponential factor.

Main Results:

  • * A general analytical approximation for metastable behavior was derived.
  • * The approximation accurately includes the pre-exponential factor in switching time calculations.
  • * The framework was successfully applied to a gene expression model exhibiting bistable switching.

Conclusions:

  • * The developed perturbation framework provides accurate approximations for metastable behavior in stochastic systems.
  • * This method offers an improvement over traditional diffusion approximations for analyzing long-time dynamics.
  • * The study highlights the utility of the framework for complex biological systems like gene expression regulation.