Thermodynamics and statistical mechanics is only applicable for a assembly of large number of particles. If you are talking about particles of 10 nm then I would say it still approximately applicable because you have still a large number of particles but if you are considering particles 1 nm or less then the applicability is dubious.

I am not so sure about the ergodic assumption. But you can break ergodicity in many large system like spin glass etc.

Any contact of the small subsystem with the thermostat will suffice for the equilibrium thermodynamics to be valid asymptotically. Inequilibrium thermodynamics is a different thing: a link with the thermostat must be considered explicitly. Speaking about ergodicity, one assumes the Puankare cycle to be destroyed by interaction with the thermostat.

Thermodynamics and statistical mechanics do apply to small physical systems but they require a large ENSEMBLE which means the physical system has to be repeated (in space or in time) many times/copies.

What about for example nanofluidics? It is not possible to make a link between small and large ensemble, and what about defining a temperature at this small scale?

I think it make sense to use thermodynamics in nanophysics. Nanostructures relate to phase, phase relates to Gibbs free energy, Gibbs free energy relates state function, and phase transitions relate to thermodynamic processes.

To answer this question we have to understand in what ways a nano system differs from macroscopic system. In my opinion the two basically differ in terms of the number of particles and in terms of its energy spectrum and surface effects. Since the thermodynamics or statistical physics are in principle applicable to large N systems, strictly speaking we can not use thermodynamics or statistical physics. However, let us examine how N plays its role. Statistically, a system of N particles can be considered as a system of N particles with an error of $ \pm \sqrt{N}$ in N, which means that if we have $10^{10}$ particles in a system, the error in N could be as large as $10^5$ indicating that the properties of the system (say average thermal energy) would have an error of about $\pm 10^{-3}$%. If your experimental arrangement is such that this error is smaller than experimentally expected error, then you can certainly use thermodynamics or statistical physics to explain your observation. It is evident that larger is the N lower would be such % errors.

However, since nano-systems have nearly $10^4$ particles, the statistical error involved is about $10^2$ indicating that the % error in the properties would be about 1%. Similarly such errors in case of a nano system of about 100 particles would be 10% and so on. This explains how disagreement between theory and experiment increases with decrease in N if it only depends on N. However, the energy spectrum of the nano-system also differs significantly from that of bulk. Hence the disagreement between theory and experiment would increase if we use spectrum of bulk. In addition since the surface effects also play their role and these effects can not be neglected in case of nano particles.

I hope this may help in understanding the extend to which we can apply thermodynamics or statistical physics. I further hope that people would correct me if there is any error in these arguments.

For nanostructures the thermodynamic limit and any type of adiabatic approximation do not apply . Strictly speaking, they are quantum objects. They exhibit quantum confinement as well as quantum fluctuations which are experimentally confirmed .

Microscopically observable properties like temperature, volume and pressure etc. would fluctuate not only more in values but also more rapidly. We may have to ask temperature in this range on such and such time. We may need to talk of transient macroscopic properties - of course time averages over several cycles may be defined but for smaller systems such variations may be significant. It is possible that we may have to shun statistical thermodynamics. New parameters to define equilibrium thermal state etc., instead of temperature may be needed. The question of probability of occupancy of a state and Boltzmann entropy .law defining equilibrium as most probable state through s=k ln(W) may need a re-look.What is equilibrium if temperature is so time dependent and pressure through so much force fluctuation dependent. But all these arguments made are spontaneous and require more thinking from me.

Vijay Jindal, I totally agree with you.

Yes it does, with some care of course. Computer simulations (based on Statistical Mechanics) have been used along the time to study small systems, including clusters of a few tens or hundreds of particles, and many results can be interpreted in terms of (pseudo-)phase transitions, including things that seem in principle purely macroscopic, like crystallization or melting.

In real application nanophysics face temperature related parameters. In nanomagnetism case strongly depend on structure and temperature. In microscopy area there are lot of thermodynamic and statistical argument exist which is related with volume. In nanophysics volume is negligible so it is difficult to use thermodynamics existing role. So I think near in future thermodynamic and statistical assumptions will important for nanophysics.

It does make sense, once physical properties of nanosystems have to be described (calculated), since most of these properties are macroscopical anyhow. My answer applies only to (macroscopic) physical properties such as Kerr rotations or electric resistances and not to an esoteric discussion of the applicability of statistical physics. By the way ln(n!) there does not have to be approximated by the Stirling formula.

A lot has been said already in other mails. What makes sense in many cases where such questions come up, and here we talk about temperature, statistical mechanics, for nanoscale systems, to go back to the root, i.e. the basics of statistical physics. There you will find, when quantities like Temperature are derived, what assumptions and approximations are taken into account.

Many people talk about temperature and other quantities where they actually do not apply as these are macroscopic or large N quantities.

...and think about why we have defined such things as the microcanonical, canonical and grand canonical ensembles in statistical physics - which quantities are defined (and how they are being defined) and which ones aren't .

It is difficult to talk about an isolated small object (as e.g. clusters mentioned above) having a specific temperature (because when isolated, it 'lives' at constant energy), but of course you could have an ensemble of such objects in (or nearly, approximately so) equilibrium with a heat bath of some temperature.

In quite a few experiments including the advanced ones on cluster melting, for example, we start out with such a quasi equilibrium ensemble but then go out of contact with the heat bath and continue with processes which subsequently need to be treated micro-canonically (absorption of laser photons, evaporation of atoms). What you need to do then, is to find the change of initial heat bath temperature which (on average) leads to the same modifications as the absorption of a photon of known energy. Such experiments were beautifully done by B. von Issendorff and colleagues. When dealing with the microcanonical ensemble, we may observe negative specific heat (at "phase transitions" resp. their microsc. analogues), which is strictly impossible in the canonical ensemble.

Similar statistical thermodynamics is done to describe the effects of energetic impact involving nuclei. There is a textbook on this kind of thermodynamics by Gross:

World Scientific Lecture Notes in Physics: Volume 66

Microcanonical Thermodynamics:

Phase Transitions in “Small” Systems

Dieter H E Gross

So yes, meaningful statistical analysis can be done for "small" systems, but you need to know what you do and why so. 'Straightforward' sqrt(N) arguments are probably misleading more often than being helpful. I agree with the statements of Robert above.

You can always define a temperature, e.g., in a microcanonical setting from the partition sum, even for a small system. So, I am no sure I agree temperature is a large N propery per se.

Kai Fauth, I am not sure but aren't you making a confusion between a nanoscale region within a macroscopical ensemble and an ensemble of isolated nanoscale objects? I have to think about that...

Very interesting question! Note that the traditional formal definition of temperature T in Stat Mech is clearly inapplicable to a small system with discrete energy levels and discrete values of the entropy S for different (also discrete) values of the total energy U. In these cases 1/T would have to be a tensor quantity instead of a simple derivative of S with respect to U. Stat Mech is a powerful way of looking at BIG systems where the discreteness can be ignored. For very small systems we can still play the Boltzmann game by postulating that the small system is "in thermal equilibrium with" a big system whose T is better defined; but this just shoves our ignorance into the mysterious "thermal equilibrium" box.... I'd love to see a frontal assault on the problem from first principles, rather than just dodging the issue.

Professor Bridgman in his classic monograph entitled "The Nature of Thermodynamics" (1969) has clearly illustrated "the universe of operations of thermodynamics". He asserts that all the measurements that in present day we can perform do not fall within the universe of operations of thermodynamics. Thermodynamics encompass only those measurements which correspond to the plateau region with respect to the size and the time duration of measurements. As the size and/or time duration of measurements decreases below the plateau region the natural fluctuations appear. These fluctuational regions are not governed by the second law of thermodynamics. There one needs to follow the probabilistic description. Thus if one is dealing with the large collections (astronomical number) of nano-particles its thermodynamics is possible provided our measurements do not fall in the fluctuational region of time duration of measurement. Even if one is not dealing a system of very small size still the measurements can fall outside the scope of thermodynamics if our time duration of measurements are of the order of a picosecond. For example, when two macroscopic systems are in thermal equilibrium, that is having same temperature, when brought in a diathermal contact and measure the energy transferred in picosecond intervals we will find the net energy transfer from one system to another even though the temperatures of two systems are same.

Yes it does! For instance, some of the interesting problems in the nanoscale are related to heat transport. See for instance Phys. Rev. B 88, 085122 (2013).

In Materials Science we do it every time. In fact, properties like surface tension and critical nuclei sizes are in the realm of nanosciences since the time of Gibbs and they were introduced to reconcile thermodynamics with the ultrasmall systems. My colleague, Byeong-Joo Lee, from Pohang has been working in this field in the last years, with success.

Please search for: Nanothermodynamics.

Nathalie, I don't think I do.

Nevertheless, the subject is subtle. As Jess correctly points out, "thermal equilibrium" is one of the key words and it may be regarded as dishonest to assume it is given without telling how this is achieved. Ideally, it is not required to know _how_ thermal equilibrium materializes if only it does. And I have worked on this basis many times. On the other hand, equilibrium is not possible without interactions, and then you do not have isolated (nano-) systems any more, strictly speaking.

My notions above were intended to address a (macroscopic, if you like) ensemble of nanoscale objects for which you assume that thermal equilibrium is given while assuming atthe same time that the identity of the nanoscale objects is unchanged (so that we are able to talk about it, actually).

Concerning the melting subject I wanted to point out that certain experiments do change the kind of ensemble along the way. This is not infrequent in molecular beam setups and thermalization in such beams is an important topic.

Kai Fauth, could you give an example of nanoscale objects which interact but keep their identity?

For a small system we can define temperature by statistical averaging. But it cannot be used in the Clausius' statement of the second law of thermodynamics. For the latter one needs a temperature based on very large lnN.

I believe that the question is related to the system having fewer particles; not with a nano-size system within a macroscopic system of particles. In the latter case there is nothing like surface effect or the effect due to difference in energy spectrum of a nano-system from bulk; of course one may think of the large fluctuations in local T, P etc. Evidently, the properties of the former would differ from the properties of the latter which would be identical to the properties of the bulk. In certain cases difference in the properties of a nano-system may be significantly large which could better be related to a phenomenon such as phase transition (possibly a pseudo).

It is true that it is not only the temperature (T) but in principle all thermodynamic quantities such as entropy, pressure, internal energy, etc. of a system of fewer particles is ill-defined due to their large fluctuations with time. However, if the system is in equilibrium with a thermal bath at a given T, statistical mechanics can be applied to the system since the time needed for measurement the $T$, pressure, etc is large enough for the particles to occupy different states as per the laws of their distribution on different energy levels and what we get is an thermodynamic average of the internal energy, P, etc. Hence I do not know if 1/T is still a tensor; perhaps Brewer may clarify this in detail. I also guess that the measurements of the properties of a nano system in the laboratory are in general performed when it is in equilibrium with a bath at constant T. As agreed the basic parameters which are involved in evaluating the properties of a nano-system have large fluctuations and they seem to question the possibility of application of statistical mechanics in deriving the thermodynamic properties of the system because the said fluctuations are likely to render errors in the process of their calculations. And in this context $\sqrt{N}$ law of fluctuation in $N$ can be a way to perceive the errors and their magnitude; here we assume that we also estimates the N (it is not a given parameter).

This is turning out to be a fun topic. I may have to go back and actually read all the postings! Just for now, though, I want to toss out a couple of remarks:

1. Stat Mech can deal with fluctuations quite handily, as long as you are able to rationalize the meaning of "thermal equilibrium" in context. (My colleague Birger Bergersen used to teach a grad course that covered this nicely; I wish I had acted upon my urge to audit it. :-)

2. If you focus on the "really really small" and "very very short times", you have no choice but to delve into quantum mechanics and ultimately quantum field theory (QFT), where even the conservation of energy has to be redefined. Processes can "embezzle from the energy bank" as long as they redeposit the borrowed energy before its absence is noticed. This is the basis of the most successful physical theories ever devised: quantum electrodynamics and QCD. So in a sense field theory deals with just the question you have asked, but not in the same language as Stat Mech. I have a gut feeling that further synthesis is needed, but maybe QFT is enough.

Jess, ok, but I talked about nanometers scale, not the Planck length!!!

There is a Density Matrix apparate in the Statistical Mechanics, which is useful in a case of description small quantum systems with a macroscopic bath around them. The temperature is deternined in this case by the temperature of bath. The Liuville -- von Neumann equation for the density mattrix allow us to describe both stationary and time-dependent cases.

@Nathalie: What, you think QED doesn't apply on the scale of nm? ;-)

As for me it is poosible and inevitable for the general functionality of the abstract probability concept. As for temperature, it exists without particles as a selfrelevation of an anergy radiation field, see Gay-Lussac experiment. The only problem is to decide properly what the phase space for the given task really is. Not the mpottion of particles determine the temperature but the temperature determins the particl motion.

As for me it is poosible and inevitable for the general functionality of the abstract probability concept. As for temperature, it exists without particles as a selfrelevation of an anergy radiation field, see Gay-Lussac experiment. The only problem is to decide properly what the phase space for the given task really is. Not the mpottion of particles determine the temperature but the temperature determins the particle motion.

Chapter Information Thermodynamics

Nathalie, your question is quite general and it id difficult to provide an answer at the same level of generality without remaining too vague.

There are a few facts one should take into account:

1. classical thermodynamics, in its usual form, heavily uses consequences of a huge number of degrees of freedom in macroscopic objects. The very first thing I can cite is that the usual definition of extensive quantities depends on the possibility of neglecting surface contributions with respect to bulk (volume) contributions. Also neglecting fluctuations is allowed only for very large systems.

2. Gibbs' statistical mechanics is formulated in term of ensembles of finite systems.

However, it is well known that only after taking the so called thermodynamic limit one can recover the usual thermodynamics. For example, different ensembles provide different results(are not equivalent) for a finite number of degrres of freedom. Moreover, it is also well known that no phase transition (defined as non analytic behavior of a thermodynamic potential) can occur in a finite system.

Nevertheless, it is possible to apply a great part of theermodynamics and statistical mechanics results to small systems, if some care in taken.

By hart I remember there should be a classical book by Hill on statistical mechanics of small systems. Moreover, in recent years there has been intensive research on things like deviations from the second principle of thermodynamics for mall systems.

A part these sparse comments, I would add another more specific to youor question about temperature of small systems.

If you refer to the average kinetic energy certainly it becomes less and less well defined when the system size decreases. However, if your small system is in equilibrium with a thermostat, the thermostat temperature, by definition of thermostat, it is a well definite object and does not depends on the size of the cluster.

Finally, as far as fluctuation-dissipation theorem is concerned , I think there has been some work to extend it to small systems but I should check in my bibliographic records to see if I can recommend some specific reading about this point.

I would think it largely does, at least for the "classical" degrees of freedom; molecular dynamics simulations over fractions of picoseconds and at nanometer sizes mostly give results consistent with equilibrium termodynamics.

Quantum stuff is probably different. In some cases, averaging (or self-averaging) does not occur even at the mesoscopic (micron) scale: think of weak localization or conductance fluctuation in Aharonov-Bohm like experiments on electron gases.

@Nathalie, coming back to your question further up: "keeping their identity" is, of course, a question of the criterion applied. To make it practical: for some purpose it might be adequate to say that a CO molecule adsorbed on, say, graphene stays a CO molecule. It is not unmodified but still retains much of its characteristic. Or take a hydrogen atom in space, interacting with the radiation of the stars...

The level of the physical theory you may use is a separate question from how statistical mechanical arguments may be applied practically. For the latter there is always a sufficiently fine-grained element of space-time that allows you to treat it as effective equilibrium. For example using a small element of 3-space containing enough particles to do your analysis with a small enough time element that few interactions take place is always possible. The question then becomes a practical one: Is the resulting numerical work so large as to exceed your computing capacity? These methods are successfully applied to large nuclei and to quark-gluon plasmas.

Natalie, perhaps it might help You a method for treatment an interaction (linear) of a quantum system with reservoir at given temperature, the Caldeira-Leggett method. For a review see Rev. Mod. Physics 59 (1987) , 1 and the literature therein. The quantum systems treated there are much simpler than Your nanosystem is, You have to find a simple adequate model according to what do You expect from Your calculation.

A collection of nanoparticles with or without interaction between them is not the same as the individual isolated nanoparticle. The real interest in nanophysics and nanotechnology lies in isolated or semi-isolated nanoparticles.The application statistical mechanics in real small nanoparticles is limited and approximate whereas statistical mechanics and thermodynamics can still be used intelligently albeit approximately for assemblies of namoparticles with weak and strong interactions.

A lot of research money has already been wasted for studying assembly of nanoparticles. This is really a very old problem of chemistry. The chemists often produced assembly of nano-size particles in chemical reactions which showed broadened x-ray diffraction peaks. They then tried to produce larger particles (bulk) for x-ray diffraction studies. Many of them did not have the possibility of using electron microscopy but some of them did use it. There was less reasearch money then. Now a huge number of material scientists are just repeating the works of the old solid state chemistry and flooding the literature with mostly uninteresting and known results.

For real nanoparticle research one should concentrate on isolated and semi-isolated particles. Here the application of traditional statistical mechanics or thermodynamics is questionable and can be done with modifications of definitions and assumptions and techniques. There are already many attempts along this directions.

I was just reading a very nice article by W.T. Coffey and Y.P. Kalmykov entitled "Thermal fluctuations of magnetic nanoparticles: Fifty years after Brown". Here it is

I am sending the refernces which touch this problem I hope that You find something

interesting The size effect in equation of state for nanostructures, K. Warda, J. Phys. Cond. Matter 21, No 34, 345301 (2009);

Tapan has not only clarified the problem but has also provided helpful reference given. It is in this context I too tried to explain (see my first post) how difference in theory and experiments increase with decrease in number of nano-particles in their ensemble

You might also be interested in this recent paper by Polak and Rubinovich:

The Intrinsic Role of Nanoconfinernent in Chemical Equilibrium: Evidence from DNA Hybridization

NANO LETTERS Volume: 13 Issue: 5 Pages: 2247-2251 DOI: 10.1021/nl4008198 Published: MAY 2013

Best

Al

The question is very interesting and makes me think about the fundamentals of the statistical mechanics. I do not have an answer to give, but I would like to make some considerations.

1- Statistical mechanics is derived from the laws of statistics with the further assumptions that all the states are equiprobable and the ergodicity assumption. Now, we can apply the statistics also to the a single dice. Moreover it is used to predict election results, the probability of car accidents and so on, which are in general small systems, comparable in size with nanoscale physical systems. In the case of statistical physics, we have used the REMC approach (see attached paper) to calculate thermodynamic quantities of a non-ideal gas. The REMC consider a nanoscale portion of the whole system sampling the canonical ensemble. I am not sure we can consider this case an example of nanoscale system, but it converges rapidly to a stationary state, with small fluctuations.

2 - To push the concept to an extreme, II have also started to think about quantum statistical mechanics applied to a single atom. In principle it is possible, because quantum statistical mechanics refer to the states of the systems and the states of the environment. The simplest system I can think about is an hydrogen atom in a box with reflecting walls. Starting with an atom outside equilibrium, excited in a given electronic state. How the system evolve? it reaches an equilibrium state, or at list it remains close to an equilibrium configuration for the large part of the time? I suppose that the atoms start to emits photons that are absorbed and reemitted many times, spreading the distribution of levels and creating a bose gas of photons. The macroscopic properties of the photon gas and of the atomic states oscillate around an equilibrium value?

More questions than answers!

Sometime ago (late nineties) we asked ourselves almost the same question in connection with ultra cold gases. We spend quite some time to find an answer that satisfied us. In my opinion the publication in Solid State Communications  Volume 109, Issue 9, 19 February 1999, Pages 615–620 summarizes our findings. As long as you can calculate the density of states and its transforms, you have access to most quantities used in equilibrium statistical mechanics. When dealing with fluctuations you should look to the last paragraph of arXiv:cond-mat/0303546 This indicates that the statistical quantities of the system in equilibrium also change under influence of an external perturbation.

http://arxiv.org/abs/cond-mat/0303546

Thermodynamics is going not be very useful at very small scales. Any estimate of temperature is going have greater uncertainty the smaller you go in scale.

If you have N particles interacting, you can still use statistical mechanics. The statistics can estimate any uncertainties in your measures. Uncertainty will increase as N goes down. For for any specific case you'll know if your numbers are useful or not.

Dear Nathalie,

The notion of temperature totally makes sense if you identify it with the temperature of the thermal reservoir your small set of particles is in equilibrium with. In this context, you will be able to prove FDT and even stronger relations, usually called fluctuation theorems, which generalize FDT and Onsager relations beyond the linear response regime. You can also think in terms of thermodynamics when dealing with processes associated to these particles, yes. But on these small scales you need a stronger machinery, because in this case fluctuations are important. This is the subject of stochastic thermodynamics, which in fact is a really hot topic nowadays, althouhg the first papers which opened this door are about 20 years old. These ideas have been succesfully applied to interpretation of experiments at the single molecule level, for example, in the estimation of energies associated to the bonds in DNA hairpins. Usually in these experiments one couples a small particle, trapped with optical tweezers, to a piece of DNA and one moves then the optical trap in a controlled way. Other succesful context for stochastic thermodynamics is associated to the physics of molecular motors, where now one can propperly define and study their efficiency. From this, and the piece of nice theoretical work I'm refering to, much valuable information is being obtained at that scales. If you would like to read some papers on the theory of stochastic thermodynamics, or on these experiments related to the micromanipulation of biomolecules and molecular motors, I can provide some references.

Best wishes,

Reinaldo.

There is no problem using Stat Mech on single particles in thermal contact with a heat reservoir at temperature T, that's what the Boltzmann distribution is for. However, defining the entropy and temperature of a small quantum system can get tricky: The possible values of the total internal energy of the system are limited & discrete, and so are the numbers of different microstates that have a given energy. Therefore we run into some problems if we define 1/T as the derivative of entropy with respect to energy, as neither of those are continuous variables. Probably the temperature will have to be reimagined as a tensor quantity of some sort. But I doubt that this makes Stat Mech "inapplicable".

Yes it makes sence if you understand Symmetry in the interaction of different brain molecules.