# statistical mechanics

## Freely jointed chain

Submitted by Zhigang Suo on Mon, 2013-02-18 20:45.A single strand of polymer is a chain of a large number of monomers. The monomers are joined by covalent bonds, and two bonded monomers may rotate relative to each other. At a finite temperature, the polymer rapidly changes from one configuration to another. When the two ends of the polymer are pulled by a force, the distance between the two ends changes. The polymer is known as an entropic spring. These notes are developed as part of statistical thermodynamics to supplement the course on advanced elasticity.

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## Open Postdoctoral Position at Carnegie Mellon University (USA)

Submitted by Radu Marculescu on Tue, 2012-12-18 02:01.We are currently looking for a top candidate to join the System Level Design group at CMU (www.ece.cmu.edu/~sld/), as a Postdoctoral Associate, starting Spring 2013. Main responsibilities involve work on modeling and control of stochastic micro-robotic swarms targeting biological applications.

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## Carnot Cycle

Submitted by Zhigang Suo on Sat, 2011-10-15 15:54.
(Class notes for ES 181 Engineering Thermodynamics. Also part of my notes on thermodynamics) In 1824, Sadi Carnot (1796-1832) published a short book, *Reflections on the Motive Power of Fire*. (The book is now free online. You should try it out.) To construct an engine, Carnot noted, at least two reservoirs of energy of different temperatures are needed. He further noted that the engine loses efficiency whenever the working fluid of the engine exchanges energy with the rest of the world across a finite difference in temperature. To avoid such exchange of energy, he described a specific cycle—later known as the Carnot cycle—consisting of isothermal and adiabatic processes. Whenever the working fluid exchanges energy with either reservoir, the temperature of the working fluid is kept the same as that of the reservoir. Whenever the temperature of working fluid differs from the temperatures of the reservoirs, the working fluid is thermally insulated. He argued that this cycle is the most efficient of all cycles that convert heat to work by operating between two constant-temperature reservoirs of energy.

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## Statistical Mechanics of Humans

Submitted by Libb Thims on Tue, 2010-12-28 05:49.This week I am working on a chronologically ordering of the 400+ thinkers of the past and present who have applied thermodynamics, whether chemical or statistical, to the humanities. In this group, germane to iMechanica, a few have specifically applied statistical mechanics to explain things such as crowd behavior, exchange of goods via money on small islands, traffic patterns, among numerous other order/disorder speculations on the microstates of society schemed on Boltzmann entropy models. I use the gas particle icon (adjacent) to distingish this group. Some of these mechanically-minded thinkers take years to discover, so I thought I would post a quick blog note here to stir up some possible discussion as to thinkers along these lines I might have missed in my list.

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## Microcanonical Entropy and Mesoscale Dislocation Mechanics and Plasticity

Submitted by Amit Acharya on Mon, 2010-11-08 21:34.(Journal of Elasticity, Carlson memorial Volume)

A methodology is devised to utilize the statistical mechanical entropy of an isolated, constrained atomistic system to define the dissipative driving-force and energetic fields in continuum thermomechanics. A thermodynamic model of dislocation mechanics is discussed. One outcome is a definition for the mesoscale back-stress tensor and the symmetric, polar dislocation density-dependent, Cauchy stress tensor from atomistic ingredients.

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## Free Energy

Submitted by Zhigang Suo on Mon, 2009-02-23 14:54.For a system in thermal contact with the rest of the world, we have described three quantities: entropy, energy, and temperature. We have also described the idea of a constraint internal to the system, and associated this constraint to an internal variable.

The system can be isolated at a particular value of energy. For such an isolated system, of all values of the internal variable, the most probable value maximizes entropy. We will paraphrase this statement under two different conditions, either when the entropy is fixed, or when the temperature is fixed. Under these conditions, the system is no longer isolated. Consequently, we need to maximize or minimize quantities other than entropy.

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## Lecture notes on "Elasticity" and "Statistical Mechanics"

Submitted by Cai Wei on Thu, 2008-06-05 20:50.The lecture notes of the two courses I taught at Stanford University during the last two quarters, "ME 340 Elasticity" and "ME 334 Introduction to Statistical Mechanics", are available in PDF format online at:

http://micro.stanford.edu/~caiwei/me340/

http://micro.stanford.edu/~caiwei/me334/

Perhaps it could be useful to you.

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## 5 Lectureships and 5 Post-Doctoral Research Fellowships -national Centre for Advanced Tribology at Southampton (nCATS)

Submitted by nt3 on Mon, 2008-05-19 06:34.

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## What is "randomness"?

Submitted by Ajit R. Jadhav on Sat, 2007-04-07 00:43.Does the word "randomness" have antonym? If yes, what is it? Why? What view of randomness does that imply?

The notion of randomness is, of course, basic to both statistical mechanics (or kinetic theory) and quantum mechanics. But these are not the only fields where it is relevant. The notion also appears virtually in any field where probabilities are used. For example, we speak of random loads and vibrations (in structures and machine design), random noise (say, in acoustics), etc. But what does the term "randomness" really mean? Any idea? What would you say? Here, I am here looking for brain storming, so half-baked ideas, side comments, etc. etc. are welcome.

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## A book on mechanics that would pique your curiosity

Submitted by Ajit R. Jadhav on Wed, 2007-03-28 01:03.I am happy to recommend the following book for your general reading.

Ranganath, G.S., ``Mysterious Motions and other Intriguing Phenomena in Physics," Hyderabad, India: Universities Press (2001)

## thermodynamics of nanoscale small systems

Submitted by Henry Tan on Thu, 2007-03-15 17:29.How to measure the temperature of a nanotube?

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## Electric potential

Submitted by Zhigang Suo on Fri, 2007-02-23 16:26.- Electric charge
- Movements of charged particles
- Elastic dielectric
- Work done by a battery and by a weight
- Electromechanical coupling
- Conservative system
- Experimental determination of electric potential
- Lagendre transformation
- parallel-plate capacitor

Return to the outline of Statistical Mechanics

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## Chemical potential

Submitted by Zhigang Suo on Thu, 2007-02-22 22:30.- A system that can exchange particles with the rest of the world
- Chemical potential
- Ideal gas
- Experimental determination of chemical potential
- Lagendre transformation
- Ideal gas once more
- Experimental determination of chemical potential
- A system in contact with a reservoir of energy, volume and particles
- A kinetic model

Return to the outline of Statistical Mechanics

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## Pressure

Submitted by Zhigang Suo on Tue, 2007-02-20 13:00.So far we have been mainly concerned with systems of a single independent variable: energy (http://imechanica.org/node/4878). We now consider a system of two independent variables: energy and volume. A thermodynamic model of the system is prescribed by entropy as a function of energy and volume.

The partial derivatives of the function give the temperature and the pressure. This fact leads to an experimental procedure to determine the function for a given system.

The laws of ideal gases and osmosis are derived. The two phenomena illustrate entropic elasticity.

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## The Boltzmann Distribution

Submitted by Zhigang Suo on Sun, 2006-10-08 16:52.- A small system in thermal contact with a large system
- The Boltzmann factor
- Partition function
- The probability for a system in thermal equilibrium with a reservoir to be in a specific state
- The probability for a system in thermal equilibrium with a reservoir to be in a configuration
- Thermal fluctuation of an RNA molecule
- A matter of words

Return to the outline of Statistical Mechanics.

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## Entropy

Submitted by Zhigang Suo on Sun, 2006-10-08 16:46.- A dissection of a sample space
- Entropy of a dissection of a sample space

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## Energy and Fundamental Postulate

Submitted by Zhigang Suo on Sun, 2006-10-08 13:20.We have described two great principles of our world: the fundamental postulate and the conservation of energy. The former is the foundation of thermodynamics, as we have learned in a previous lecture. The latter is not specific to thermodynamics: we borrow the concept of energy—along with the principle of the conservation of energy—from other branches of science, such as mechanics and electrodynamics. Both principles are abstracted from many empirical observations.

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## Fundamental postulate. Entropy

Submitted by Zhigang Suo on Sun, 2006-10-08 13:15.Of our world the following facts are known:

- An isolated system has a set of quantum states.
- The isolated system ceaselessly flips from one quantum state to another.
- A system isolated for a long time is equally probable to be in any one of its quantum states.

Thus, an isolated system behaves like a fair die. The following notes remind you what an isolated system is, and translate the theory of probability of rolling a fair die to the thermodynamics of an isolated system.

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## Probability

Submitted by Zhigang Suo on Sun, 2006-10-08 13:10.- An experiment that has many possible outcomes
- Construct a sample space at a suitable level of detail
- Probability of an event
- Conditioning
- Independent events
- Random variable
- Use a random variable to specify an event
- Use a random variable to dissect a sample space
- Probability distribution of a random variable
- Variance of a random variable
- A dimensionless measure of the fluctuation of a random variable

Return to the outline of Statistical Mechanics

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## Statistical Mechanics

Submitted by Zhigang Suo on Sun, 2006-10-08 13:01.Here are sections that I have now:

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## A Fresh Look at a Beautiful Subject

Submitted by Zhigang Suo on Mon, 2006-09-18 14:37.This is a review on *Thermal Physics* by Charles Kittle and Herbert Kroemer. I posted the review on Amazon on 2 December 2001.

This is by far THE BEST textbook on the subject. As many people say, thermodynamics is a subject that one has to learn at least three times. I can easily understand the very negative review from the undergraduate student at Berkely. The subject itself is hard, and simply is not for everyone, not for the first run at least. I say this from experience. I earned a Ph.D. degree over ten years ago, and took courses on thermodynamics at both undergraduate and graduate levels. I didn't understand the subject at all, and didn't find much use in my thesis work. However, something about the subject has kept me going back to it ever since. I now own about 40 books on the subject, and use the ideas almost daily in my research.

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