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Second Order systems II

Second Order systems II

In an underdamped second-order system, where the damping ratio ζ is between 0 and 1, a unit-step input results in a transfer function that, when transformed using the inverse Laplace method, reveals the output response. The output exhibits a damped sinusoidal oscillation, and the difference between the input and output is termed the error signal. This error signal also demonstrates damped oscillatory behavior. Eventually, as the system reaches a steady state, the error diminishes to zero.

First Order Systems

First Order Systems

First-order systems, such as RC circuits, are foundational in understanding dynamic systems due to their straightforward input-output relationship. Analyzing their responses to different input functions under zero initial conditions reveals significant insights into system behavior.
When a first-order system is subjected to a unit-step input, its response is characterized by its transfer function. By applying the Laplace transform of the unit-step input to the transfer function, expanding the...

Second Order systems I

Second Order systems I

A servo system exemplifies a second-order system, featuring a proportional controller and load elements that ensure the output position aligns with the input position. The relationship between these components is described by a second-order differential equation. Applying the Laplace transform under zero initial conditions yields the transfer function, showing how inputs are converted to outputs in the system.
By reinterpreting the system, one can derive the closed-loop transfer function, which...

Thermodynamic Systems

Thermodynamic Systems

A thermodynamic system is a set of objects whose thermodynamic properties are of interest. The system is considered to be embedded in its surroundings or the environment. The system and its environment can exchange heat and do work on each other through a boundary that separates them. However, the immediate surroundings of the system interact with it directly and therefore have a much stronger influence on its behavior and properties.
Consider an example of tea boiling in a kettle. The...

Classification of Systems-I

Classification of Systems-I

Linearity is a system property characterized by a direct input-output relationship, combining homogeneity and additivity.
Homogeneity dictates that if an input x(t) is multiplied by a constant c, the output y(t) is multiplied by the same constant. Mathematically, this is expressed as:

Classification of Systems-II

Classification of Systems-II

Continuous-time systems have continuous input and output signals, with time measured continuously. These systems are generally defined by differential or algebraic equations. For instance, in an RC circuit, the relationship between input and output voltage is expressed through a differential equation derived from Ohm's law and the capacitor relation,

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Related Experiment Video

Updated: Feb 10, 2026

A Quantitative Fluorescence Microscopy-based Single Liposome Assay for Detecting the Compositional Inhomogeneity Between Individual Liposomes

A Quantitative Fluorescence Microscopy-based Single Liposome Assay for Detecting the Compositional Inhomogeneity Between Individual Liposomes

Published on: December 13, 2019

Pseudochemotaxis in inhomogeneous active Brownian systems.

Hidde D Vuijk¹, Abhinav Sharma¹, Debasish Mondal^1,2

¹Leibniz-Institut für Polymerforschung Dresden, Institut Theorie der Polymere, 01069 Dresden, Germany.

Physical Review. E

|May 16, 2018

Summary

This study investigates how inhomogeneous activity affects self-propelled Brownian particles. Optimizing activity profiles significantly enhances target-finding probability and reduces travel time, mimicking chemotaxis.

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Published on: June 12, 2015

Area of Science:

Physics, Soft Matter
Statistical Mechanics

Background:

Self-propelled Brownian particles exhibit complex dynamics influenced by their environment.
Understanding particle behavior in inhomogeneous fields is crucial for applications in micro-robotics and biological systems.

Purpose of the Study:

To investigate the dynamical properties of confined, self-propelled Brownian particles under inhomogeneous activity.
To quantify the impact of activity profiles on target-reaching probability and mean first passage time.
To explore the emergence of directed motion (chemotaxis-like behavior) due to activity gradients.

Main Methods:

Brownian dynamics simulations were employed to model particle trajectories.
Calculations included target-reaching probability and mean first passage time.
An approximate Fokker-Planck equation was derived by integrating out orientational degrees of freedom.

Main Results:

Inhomogeneous activity profiles significantly influence particle dynamics.
Placing high-activity zones between the start and target enhances target-finding probability and decreases passage time.
Activity gradients induce a drift, causing particles to move up the gradient, similar to chemotaxis.

Conclusions:

The spatial distribution of activity is a critical factor in controlling the collective behavior of active Brownian particles.
Theoretical predictions derived from the Fokker-Planck equation closely match simulation results.
This work provides insights into designing active matter systems for targeted delivery and controlled transport.