Research Projects

I am interested in the Equilibrium and Non-Equilibrium properties of Physical systems from a Statistical Mechanics perspective. I work in two areas of Statistical Mechanics: (i)Noise-induced phenomena, and (ii)Physics of nucleation.

(i)Noise-induced phenomena - Noise, while generally seen as having a destructive effect, can sometimes play a constructive role such as in the movement of tiny motors in the nanoscale world, known as Brownian motors. At these small length scales, classical thermodynamics fails to make any useful prediction and the behavior of such stochastic systems has to be understood in the framework of Stochastic Thermodynamics. In our research group, we apply theoretical approaches from Stochastic methods as well as Computer simulations to investigate these systems.

Fig: Schematic of a Brownian Motor driven by spatially non-uniform Temperature (T 1 > T2) and subject to a piecewise linear potential of barrier height U0.

During my doctoral research, I investigated the thermodynamics of a Brownian motor driven by spatially inhomogeneous temperature, known as the Buettiker-Landauer motor. Such a motor can also work as a microscopic heat pump or heat engine when it works against an external load. In general, noise or stochastic phenomena, is studied by the help of the Langevin and Fokker-Planck equations. In the high-friction limit, one usually omits the inertial term from the Langevin equation since the dynamics become overdamped. However, my investigations showed that one has to take the inertial term into account to determine the heat flows. I proved this by comparing numerical and analytical results from the Langevin equation with Molecular Dynamics simulations. My work was cited and commented upon by Prof. Ken Sekimoto, a pioneer in the field of Brownian motors and stochastic phenomena, in his book “Stochastic Energetics” (Springer, 2010). A future work involves looking for models with better efficiency.

Presently, I am working on the transport and energetics of a Brownian particle in a rough substrate, which has implications for diffusion processes in systems with a rough energy landscape such as proteins. Another topic I am investigating is the Brownian motor phenomenon involving a self-propelled Brownian particle known as active matter ratchets. Studies on such systems will help design particle separation devices at the nanoscale, transporting microscopic particles in presence of obstacles, sorting swimming Bacteria, etc.

An important area of study in the investigation of noise induced phenomenon is the effect of interaction between many Brownian particles. Such collective effects show interesting behavior such as phase transitions, spontaneous ratchet effect and negative mobility. A future project is to investigae collective effects generated by interacting Brownian particles on stochastic phenomenon. Other lines of investigation are the role of inertia of the Brownian particle and the effect of non-thermal or non-markovian noise on such phenomenon.

1.)Noise-Induced phenomena

This is a huge field. Some of the topics I am interested in are:

i.)Brownian Motors – These are tiny devices (typically in the micro-nanoscale regime) that move unidirectionally in the absence of any bias under non-equilibrium conditions and in the presence of thermal fluctuations.

References:-

(a) Brownian motors: noisy transport far from equilibrium

Link:  https://arxiv.org/abs/cond-mat/0010237

This is a very good review article. Also check out papers which have cited this article on Google scholar. Here is the link:

https://scholar.google.com/scholarcites=12374834247118906482&as_sdt=2005&sciodt=0,5&hl=en

There are actually many excellent review articles citing this paper, where you can get more information:-

(b)Artificial Brownian motors: Controlling transport on the nanoscale, Peter Hänggi and Fabio Marchesoni, Rev. Mod. Phys. 81, 387

(c)Brownian Motors, R. D. Astumian and Peter Haenggi, Physics Today 55, 11, 33 (2002)

(d) There are also very useful references in the wikipedia page of Brownian Motors: https://en.wikipedia.org/wiki/Brownian_motor

ii)Noise enhanced stability, resonant activation, stochastic resonance

Noise enhanced Stability – Consider a system in a metastable state. With the passage of time the system will explore the entire phase space and settle down to a state which is globally the most stable. However, in certain situations, such as when the system is drive by a time-periodic external force, thermal noise can ensure that the system remains in the original metastable state for a longer period of time than it would in the absence of noise. This phenomenon is known as noise-enhanced stability.

Resonant Activation – Resonant activation refers to the enhancement of the escape rate of crossing a potential barrier by a Brownian particle due to mutual interplay of a time-dependent force modulating the potential and thermal noise.

Stochastic Resonance- The enhancement of a feeble signal due to the presence of noise and external time-dependent force refers to Stochastic resonance.

Stochastic Resetting – This refers to stopping a system from reaching an equilibrium state by resetting its position continually to the initial position.

All these phenomenon have lots of technological applications and at the same time are of fundamental interest in understanding non-equilibrium statistical mechanics.

References:-

(a)Noise Enhanced Stability in an Unstable System, by Rosario N. Mantegna and Bernardo Spagnolo, Phys. Rev. Lett. 76, 563, 1996

(b)Resonant activation over a fluctuating barrier, by Charles R. Doering and Jonathan C. Gadoua, Phys. Rev. Lett. 69, 2318, 1992

(c)Stochastic resonance by Luca Gammaitoni, Peter Hänggi, Peter Jung, and Fabio Marchesoni, Rev. Mod. Phys. 70, 223, 1998

(d)Stochastic resetting and applications by Martin Evans, Satya Majumdar and Gregory Schehr, Journal of Physics A: Mathematical and Theoretical, 53193001

Books to learn more about Noise-induced Phenomena or the Physics of Stochastic Processes (The terms noise-induced and stochastic are used interchangeably)

You can look at the following books to get a broad overview of the physics and mathematics behind stochastic processes:

a)Selected Papers on Noise and Stochastic Processes by Nelson Wax. This book contains a very good collection of papers on stochastic processes. Check out the paper by S. Chandrasekhar which is the first paper one reads in order to start working in this field.

b)Fokker-Planck equation by Risken

c)Stochastic Processes in Physics and Chemistry by Van Kampen

d)Stochastic methods by Gardiner

e)Noise induced transitions by W. Horsthemke

f)Langevin Equation by Coffey

g)For a Master level introduction to Brownian motion you can refer to the Statistical mechanics books by (i)Pathria, (ii)Reif and (iii) Tuckerman

To do some path breaking research in this field you need to develop strong mathematical skills as well. You must familiarize yourself with stochastic calculus which are described in detail in the above mentioned books.

Below I am listing more resources (mainly papers) on the topic of Brownian Motors:

The following article by S. Chandrasekhar is also very useful: https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.15.1

I already mentioned this earlier.

Also look for articles in the following lecture notes in physics: Stochastic Dynamics (Editors: Lutz Schimansky Geier and Thorsten Poeschel)

https://link.springer.com/book/10.1007/BFb0105592  . There are many interesting articles in this book on various aspects of stochastic processes.

For simulations of the Langevin equation, the following articles will be helpful: 

(i)Stochastic Effects in Physical Systems by Raul Toral - https://link.springer.com/chapter/10.1007/978-94-011-4247-2_2

ii.)Numerical integration of the Langevin equation: Monte Carlo simulation by Ermack and Buckholz. Here is the link: https://www.sciencedirect.com/science/article/pii/0021999180900844

iii)A simple and effective Verlet-type algorithm for simulating Langevin dynamics. Link: https://www.tandfonline.com/doi/full/10.1080/00268976.2012.760055

iv)On stochastic approaches of nuclear dynamics.  Link: https://www.sciencedirect.com/science/article/abs/pii/0370157396000038

v) LANGEVIN APPROACH TO NUCLEAR DISSIPATIVE DYNAMICS. Link: https://hal.archives-ouvertes.fr/jpa-00225802/document

For simulation of Fokker-Planck Equation you can use finite-difference methods for partial differential equation pertaining to open or periodic boundary conditions. The Crank-Nicholson method is the best approach. Note that there is a slight difference in the algorithm for open and periodic boundary condition. You can find this algorithm in most books on numerical methods.

Look for articles citing the above papers and books, in Google scholar and Web of Science. Some of the articles are quite old (almost twenty years) so you need to update yourself on more recent works by searching for citations to the above mentioned papers.

2.)Active Matter

A topic of immense interest in the past few years in the field of Statistical Mechanics is that of Active Matter which covers a diverse group of systems such as a bird flocks, schools of fish, bacteria, artificial self propelled or active Brownian particles etc. Such active particles generate directed motion by its interaction with the environment.

(i)Link to collection of articles on Active matter in journals published by nature publishing group:

https://go.nature.com/activematter

(ii)Article Title: Simulation of the active Brownian motion of a microswimmer

Journal: American Journal of Physics

Link: https://doi.org/10.1119/1.4870398

This is a pedagogic article on simulating active Brownian particles. This is a good starting point for research on active brownian particles, especially simulations

(iii)Article Title: Computational models for active matterLink: https://arxiv.org/abs/1910.02528

As the name indicates it is an article describing various models to investigate active matter

(iv)Article Title: Novel Type of Phase Transition in a System of Self-Driven Particles

Link: https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.75.1226

This is one of the seminal papers on active matter phenomena which is also highly cited

(v)You can find other resources on the wikipedia page:

https://en.wikipedia.org/wiki/Active_matter

Check out the review article by Sriram Ramaswamy

(vi) Article Title: Active Particles in Complex and Crowded Environments

Link: https://arxiv.org/abs/1602.00081

This is a good review article on interaction of active particles with the environment. You can

also reproduce some results.

(vii)Article Title: Active Brownian Motion Models and Applications to Ratchets

Link:https://arxiv.org/abs/0801.4838

This is an excellent review article and you can also think of reproducing the results in the article. Since active matter motion is also a stochastic phenomena you should also refer to books and articles mentioned above for the topic Noise-induced Phenomena to understand more about Brownian motion and numerical/analytical solution of Langevin Equation.

3.)Nucleation, crystallization and phase behavior of various systems

Understanding of nucleation, crystallization and phase behavior of various systems in equilibrium as well as non-equilibrium conditions is one of the fundamental problems in Statistical Mechanics.

Here are some relevant references.

References:-

The following two PhD thesis are very helpful for an introduction to this topic.

i.)https://amolf.nl/publications/quantitative-prediction-of-crystal-nucleation-rates-for-spherical-

colloids-a-computational-study

ii.)Numerical Study of Pathways for Homogeneous Nucleation

(just click on above link to get the web address and then click on it)

iii.)Article Title: Homogeneous and heterogeneous nucleation of Lennard-Jones liquids

Link: https://journals.aps.org/pre/abstract/10.1103/PhysRevE.76.031604

This paper could be a good starting point to pursue research in this area.

iv)Book: Phase transformation in metals and alloys by David Porter

v)Review Article: A Review of Classical and Nonclassical Nucleation Theories

Link: https://pubs.acs.org/doi/pdf/10.1021/acs.cgd.6b00794

vi)Review Article: Crystal Nucleation in Liquids: Open Questions and Future Challenges in Molecular Dynamics Simulations

Link:https://pubs.acs.org/doi/10.1021/acs.chemrev.5b00744

The two theses and article also contain numerous references including review articles. You can read them as well.

One of the important areas if research in this field is to determine the phase diagram of various substances. For topics related to determining phase coexistence curves using free-energy based methods the following books on Computational Physics will be useful:

a.)Understanding Molecular Simulation by Frenkel and Smit

b.)Computer Simulation of Liquids by Allen and Tildesley

c.)Introduction to Computer Simulation methods by Gould, Tobochnik and Christian

d.)Computational Physics by Giordano and Nakanishi

You can look at the projects discussed at the end of various chapters. Also check out various Computational Physics Books by (i)Joel franklin, (ii)P. Scherer, (iii)Tao Pang, (iv)Stickler and Schachinger etc.

These books (1-4 above) may also be helpful for the previous topics such as active matter and noise-induced phenomena.

4.)Polymer translocation

The phenomenon of Polymer translocation is a topic of great interest in the field of Soft matter Physics.

References:-

i.)Langevin dynamics simulations of polymer translocation through nanopores

Link:https://aip.scitation.org/doi/full/10.1063/1.2357118

ii)Polymer translocation through a nanopore: A two-dimensional Monte Carlo study

Link:https://aip.scitation.org/doi/full/10.1063/1.2161189

Also check out the references in the above two articles and papers which cite this article in google scholar

iii)Review Article:Polymer translocation: the first two decades and the recent diversification

Link:https://pubs.rsc.org/en/content/articlelanding/2014/sm/c4sm01819b#!divAbstract

iv)Book: Polymer translocation by M. Muthukumar

For Monte Carlo simulation pertaining to polymers and in general for simulating, liquids, gases, crystals and Ising Model, you can refer to the books on Monte Carlo Simulations by (i)Newman, (ii)Binder and Landau

5.)Miscellaneous Topics: Non-linear Dynamics and Chaos, Random Walks

A few interesting projects in Non-Linear Dynamics and Chaos would be

(i)the investigation of chaotic ratchets (Reference:

https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.84.258)

Look for citations to this article in google scholar

(ii)Kuramoto oscillator (Reference:https://en.wikipedia.org/wiki/Kuramoto_model)

Look for references in this wikipedia page

(iii)Molecular Dynamics simulation of synchronization (Reference - Molecular dynamics simulation of synchronization of a driven particle, Tiare Guerrero, Danielle McDermott, Am. J. Phys. 89, 975–981 (2021)

For the topics: random walks, one can look at projects in the following Books. These books will also be helpful for an introduction to Non-Linear Dynamics and Chaos and interesting projects on the topic:

1.)Introduction to Computer Simulation methods by Gould, Tobochnik and Christian

2.)Computational Physics by Giordano and Nakanishi

Refer to various Computational Physics Books by (i)Joel franklin, (ii)P. Scherer, (iii)Tao Pang, (iv)Stickler and Schachinger etc.

6.)Other Topics on Soft Matter Systems

Liquid Crystals, Colloidal systems and Self-Assembly

References:-

a)Soft Matter Physics: An Introduction (Partially Ordered Systems)

by J. Friedel,Maurice Kleman, et al.

b)Soft Condensed Matter: (Oxford Master Series in Physics)

by Richard A.L. Jones

Journals:-

On all the above topics 1-6, the following journals should be looked at on a regular basis:

1)Physical Review E

2)Journal of Chemical Physics

3)Soft Matter

4)Physical Review Letters

5)JACS

6)Molecular Simulation

7)Journal of Statistical Mechanics

8)Journal of Statistical Physics

9)Fluctuation and Noise Letters

10)Physica A

11)Journal of Physical Chemistry B

12)Nature Scientific Reports

13.)International Journal of Modern Physics B

14.)European Physical Journal B

and various others.......