What is absolute temperature?

The concept and reality of absolute temperature is amazing. It tells us something fundamental about the universe, including physical limits as to what is possible. The existence of absolute temperature is intimately connected with the existence of entropy as a thermodynamic state function. It also hints at the underlying quantum nature of reality.

Aside: Unfortunately, the Wikipedia page on this topic is mediocre and garbled. For example, it continues the myth that temperature is related to kinetic energy.

The zeroth law of thermodynamics allows the definition of empirical temperature. It is an equilibrium state variable that indicates whether a thermodynamic system will remain in the same state upon being brought into thermal contact with another system. Thermometers are systems with a single state variable.

Absolute temperature is a specific temperature scale that is central to thermodynamics and statistical mechanics. 

There are several equivalent definitions of absolute temperature. They start at different points. Except for the first one, the others show that the existence of absolute temperature is intimately connected to the second law and to entropy being an extensive quantity.

This is nicely discussed by Zemansky in chapter 8 of his text Heat and Thermodynamics, Fifth Edition (1968). [This was the text for my second year undergrad thermo course at ANU in 1980. At the time, I did not fully appreciate how profound some of it is. I just enjoyed all the multivariable calculus.] 

1. Ideal gas thermometers.

Consider a fixed mass of ideal gas whose volume is fixed. An ideal gas is defined as any gas at a temperature and pressure much larger than the critical temperature and pressure for the gas-liquid transition. Suppose the system is cooled and heated, and the pressure is measured as a function of the temperature measured by a separate thermometer calibrated by the Celsius scale. The pressure versus temperature curve is a straight line. If this line is extrapolated to zero pressure, this occurs at -273.15 degrees Celsius. The straight line has different slopes for different gases, but they all intercept the x-axis at the same point. Alternatively, one can take the pressure as fixed and measure the volume of the gas versus temperature. Extrapolation to zero volume also occurs at -273.15 degrees. 

This suggests that something special is happening at -273.15 degrees Celsius. One can define a special temperature scale where this temperature is zero. Historically, this was the beginning of the concept of absolute temperature.

However, we should be cautious about this approach. This is just an extrapolation and does not allow for the fact that ideal gases are rather special or that some very different physics might kick in below the critical temperature of helium.

2. The efficiency of Carnot cycles. 

This follows Zemansky (page 208). Consider a Carnot cycle abcda, where b to c and d to a are isothermal processes, between the same two reversible adiabatic surfaces, and involve heat transfers Q and Q_3, respectively. The absolute temperature scale T is defined by 

T/T_3 = Q/Q_3

with T_3 = 273.16, when the process d to a occurs at the triple point of water.

3. Integrating factor for heat

Heat is not a state property. It depends on processes. The first law says Delta Q = Delta U + P Delta V. If we consider a quasi-static process and integrate the heat transfer along the path taken (in state space), the result may depend on the path taken. On the other hand, if one integrates dQ/T, one finds that the result is independent of the path. This can then be used to define a new state variable, the entropy. 

The brief discussion above misses some subtle and profound features that only became clear in the 1960s following the work of Pippard, Turner, Landsberg, and Sears, which was inspired by an axiomatic approach to thermodynamics developed by Caratheodory.

Zemansky states

It is an extraordinary circumstance that not only does an integrating factor exist for the dQ of any system, but this integrating factor is a function of temperature only and is the same function for all systems! This universal character enables us to define an absolute temperature.

4. Applying the second law to a composite system

This treatment follows Schroeder, Thermal Physics (Section 3.1)

Schroeder defines entropy in terms of a multiplicity of states. However, I prefer to define entropy as the state function which tells us whether or not two states are accessible from one another by an adiabatic process. There are multiple possible versions of this empirical entropy state function, but let's choose one that is extensive, i.e., scales with the mass and volume of the system.

Consider an adiabatically isolated system containing an internal partition through which the conduction of heat can occur. Denote the two parts of the system by A and B. The entropy of each part can be written as a function U of its internal energy. 

The total entropy of the system can be written 

S = S_A (U_A) + S_B (U_B)

If the system is in thermal equilibrium, by the second law, the entropy of the whole system must be a minimum as a function of U_A and U_B.

Now, dU_A = - dU_B as the composite system is adiabatically isolated. Hence, we have.

The left-hand (right-hand) side of the equation only depends on the properties of system A (B). Thus, it is an intensive state variable which determines whether the system will be in equilibrium with another system. Hence, by the zeroth law, it defines a temperature scale.

T is the absolute temperature.

What is temperature?

Temperature is NOT the average kinetic energy.

When I taught thermodynamics to second year undergraduates one of the preconceived notions that was hard to dislodge from students was that temperature IS a measure of the average kinetic energy of the atoms or molecules in a system.

First, I will give the merits of this view and then explain why it is problematic.

A profound and important insight from Maxwell's kinetic theory of ideal gases was that the average kinetic energy of the atoms/molecules in the gas is related to the absolute temperature defined by Kelvin. This result was important because it provided a microscopic basis for Joule's discovery of the mechanical equivalence of heat.

The result does not just hold for an ideal gas. Classical statistical mechanics can be used to show that for any system of interacting particles, the average kinetic energy of each particle is 3/2 kT. The proof proceeds in the same manner as the equipartition theorem. In the partition function, the integral over momentum factorises and can be evaluated exactly as it is Gaussian integral.

However, this simple relationship between temperature and kinetic energy does not hold for quantum systems. Consider the case of a harmonic oscillator, with frequency omega. By the virial theorem, the average kinetic energy is equal to the average potential energy. Thus, the average kinetic energy is half of the internal energy U(T), which is a universal function f(T/omega). Thus, if we compare two oscillators with different frequencies, at the same temperature, they will have different kinetic energies.

This problem is not just some quantum exotica that is only relevant at extremely low temperatures. Most solids are "quantum" at room temperature because they have a Debye temperature in the range of 200-1000 K.

Temperature is a macroscopic variable, not a microscopic one. It should be defined in terms of the zeroth law of thermodynamics.

Temperature is a state variable associated with a system in thermal equilibrium. It tells us whether that system will be in thermal equilibrium with another system. Consider two separated systems with temperatures T1 and T2. If they are brought into thermal contact, their states will not change if and only if T1=T2.

A thermometer is a system with a single state variable. The value of that variable is an empirical temperature.

Aside. This view of temperature was used by Planck in his book, Treatise on Thermodynamics, first published in 1905.

I am thankful to my undergraduate mentor, Hans Buchdahl for teaching me that thermodynamics is conceptually coherent and beautiful. 

This discussion illustrates that temperature is an emergent property. It is a property of a macroscopic system that the parts of the system do not have. The temperature is independent of the microscopic composition of the system or its history. This universality is a characteristic of many emergent properties.

In another post, I hope to explain what the absolute temperature, first introduced by Kelvin, is.

Maxwell's demon and the history of the second law of thermodynamics

I recently reread Warmth Disperses and Time Passes: The History of Heat by Hans Christian von Baeyer

As a popular book, it provides a beautiful and enthralling account of the discovery of the first and second laws of thermodynamics. The book is a great companion to teaching and learning thermodynamics and statistical mechanics. The narrative is unified by the puzzle of Maxwell's demon.

Aside: The book was first published in 1998 with the title Maxwell's Demon. My guess is that the publisher changed the title because most people have probably not heard of the demon, unlike Schrodinger's cat.

Baeyer captures both the wonder of the subject and the fascinating story of how the science of thermodynamics developed. He describes quirky personalities and illustrates how science proceeds with a mixture of brilliant insights, clever experiments, false leads, and forgotten discoveries. It is easy and compelling reading.

I appreciated that there is a lack of hype, in contrast to too many popular science books.

The book is enhanced by showing that the story is not over. Many reports of the demise of the demon have been premature. The penultimate chapter discusses Zurek's definition of entropy in terms of algorithmic randomness. The last chapter considers molecular motors, such as kinesin, which can be viewed as ratchets driven by thermal noise.

Physical insights

The first and second laws tell us something about the fundamental nature of the universe. Although they are macroscopic and may have some (debatable) microscopic justification,  they can be viewed as fundamental.

Central to the development of the first law was the notion of the mechanical equivalent of heat.

There are three rather different ways to formulate the second law: a Carnot cycle represents an engine of optimal efficiency, heat never passes from a cold to a hot body, and the arrow of time. It is profound that these formulations are equivalent and not something that was anticipated. We should marvel at this.

Entropy can be viewed as the absence of information. Consequently, the second law can be viewed as statistical.

Things I want to understand

A good book stimulates us to want to engage more with its subject. Some things I want to understand are the entropy of the initial state of the universe, Boltzmann's H theorem, Feynman's ratchet, Shannon's information theory, molecular motors, Zurek's definition of entropy, and Gerald Holton's book, Thematic origins of scientific thought.

A recent tutorial is A Friendly Guide to Exorcising Maxwell’s Demon, by A. de Oliveira Junior, Jonatan Bohr Brask, and Rafael Chaves

Beautiful things missed

As a popular book, I think the length and scope of topics are right. Nevertheless, in a longer book, here are some things I would enjoy reading about: the zeroth and third laws, the contributions of Gibbs, the ergodic hypothesis, Brownian motion and evidence for atoms, the role of thermodynamics (and statistical mechanics) in the development of quantum theory (blackbody radiation, Einstein solid, identical particle statistics, and the Sackur-Tetrode equation) and perhaps phase transitions.

Two quibbles

von Baeyer has a somewhat reductionist perspective that the true nature of thermodynamics was revealed by the microscopic descriptions of Maxwell and Boltzmann.

I will write separate posts on why I am not comfortable with the following two statements.

Temperature IS the average kinetic energy of molecules.

Entropy was mysterious until Boltzmann's definition S=k ln W. 

Thermodynamics and emergence

Novelty. 

Temperature and entropy are emergent properties. Classically, they are defined by the zeroth and second laws of thermodynamics, respectively. The individual particles that make up a system in thermodynamic equilibrium do not have these properties. Kadanoff provided an example illustrating the qualitative difference between macro- and micro-perspectives. He pointed out how deterministic behaviour can emerge at the macroscale from stochastic behaviour at the microscale. The many individual molecules in a dilute gas can be viewed as undergoing stochastic motion. However, collectively they are described by an equation of state such as the ideal gas law.

 Primas gave a technical argument, involving C* algebras, that temperature is emergent: it belongs to an algebra of contextual observables but not to the algebra of intrinsic observables.44 Following this perspective, Bishop argued that temperature and the chemical potential are (contextually) emergent.

Intra-stratum closure. 

The laws of thermodynamics, the equations of thermodynamics (such as TdS = dU + pdV), and state functions such as S(U,V), provide a complete description of processes involving equilibrium states. A knowledge of microscopic details, such as the atomic constituents or forces of interaction, is not necessary for the description.

Irreducibility. 

A common view is that thermodynamics can be derived from statistical mechanics. However, this is contentious. David Deutsch claimed that the second law of thermodynamics is an “emergent law”: it cannot be derived from microscopic laws, like the principle of testability.

Lieb and Yngvason stated that the derivation from statistical mechanics of the law of entropy increase “is a goal that has so far eluded the deepest thinkers.”  In contrast, Weinberg claimed that Maxwell, Boltzmann, and Gibbs “showed that the principles of thermodynamics could in fact be deduced mathematically, by an analysis of the probabilities of different configurations… Nevertheless, even though thermodynamics has been explained in terms of particles and forces, it continues to deal with emergent concepts like temperature and entropy that lose all meaning on the level of individual particles.” (Dreams of A Final Theory, pages 40-41)

I agree that thermodynamic properties (e.g., equations of state, the temperature dependence of heat capacity, and phase transitions) can be deduced from statistical mechanics. However, thermodynamic principles, such as the second law, are not thermodynamic properties. Furthermore, these thermodynamic principles are required to justify the equations of statistical mechanics, such as the partition function, that are used to calculate thermodynamic properties. 

Macro hints of microscopics.

The Sackur-Tetrode equation for the entropy of an ideal gas hinted at the quantisation of phase space. The Gibbs paradox hinted that fundamental particles are indistinguishable. The third law of thermodynamics hints at quantum degeneracy.

Introducing phase transitions to a layperson

I have written a first draft of a chapter introducing phase diagrams and phase transitions to a layperson. I welcome any comments and suggestions. Feel free to try it out on your aunt or uncle!

Does a temperature dependent Hamiltonian make sense?

At the fundamental level, we think of a Hamiltonian as independent of temperature. It is describing the energy of all possible states of the system in the absence of any environment.

However, when one does mean-field theory (e.g. for an Ising model or BCS theory) the Hamiltonian involves temperature-dependent parameters that are determined self consistently.

I have been thinking about this because one of the proposed effective minimal Hamiltonians for spin crossover compounds is an Ising model with a temperature dependent field.

My immediate reaction was that this must be some sort of mean-field theory.
However, I now realise that is not the case.

Effective Hamiltonians can be temperature dependent without invoking any approximations. Temperature-dependent interactions can arise when one integrates out some degrees of freedom.

One can see this by simply considering the case of a system with two degrees of freedom x and q. The partition function can be written as a path integral where there is an action which involves the integral of the Lagrangian in imaginary time from 0 to 1/T where T is the temperature.
Integrating out x one obtains an effective action for q that will depend on temperature.

Here are three cases where this can be done explicitly.

1. The spin boson model. One integrates out the harmonic oscillators, leading to a ``Feynman-Vernon influence functional'' that is temperature dependent.

2. A two-state system in which each state has a series of sub-states (e.g. spin states or vibrational states). Consider the simple Hamiltonian.

This corresponds to the case of spin-crossover systems and one sees how one can end up with an Ising type model with a "field" that is related to the free energy difference between the two spin states.

3. A one-dimensional chain of spin-crossover molecules which have an elastic interaction that depends on the spin state. This is treated in
Elastic interaction among transition metals in one-dimensional spin-crossover solids 
K. Boukheddaden, S. Miyashita, and M. Nishino

The classical phonons are integrated out and one is left with an Ising chain of pseudo-spins in an external ``field'' where the "exchange" interaction and field depend on temperature.
[See equation (13) in the paper].

A fascinating thermodynamics demonstration: the drinking bird

I am currently helping teach a second year undergraduate course Thermodynamics and Condensed Matter Physics. For the first time I am helping out in some of the lab sessions. Two of the experiments are based on the drinking bird.



This illustrates two important topics: heat engines and liquid-vapour equilibria.

Here are a few observations fo in random order.

* I still find it fascinating to watch. Why isn't it a perpetual motion machine?

* Several more surprising things are:
a. it operates on such a small temperature difference,
b. that there is a temperature difference between the head and bulb,
c. it is so sensitive to perturbations such as warming with your fingers or changes in humidity.

* It took me quite a while to understand what is going on, which makes me wonder about the students doing the lab. How much are they following the recipe and saying the mantra...

* I try to encourage the students to think critically and scientifically about what is going on, asking some basic questions, such as "How do you know the head is cooler than the bulb? What experiment can you do right now to test your hypothesis? How can you test whether evaporative cooling is responsible for cooling the head?" Such an approach is briefly described in this old paper.

* Understanding and approximately quantifying the temperature of the head involves the concept of humidity, wet-bulb temperature and a psychometric chart. Again I find this challenging.

* This lab is a great example of how you don't necessarily need a lot of money and fancy equipment to teach a lot of important science and skills.

The challenge of non-equilibrium thermodynamics

This week I am in Telluride at the bi-annual workshop on Condensed Phase Dynamics. I really enjoyed the talks today. A common topic was that of non-equilibrium thermodynamics, particularly in nanoscale systems.

Abe Nitzan began his talk mentioning a recent PRL, Quantum Thermodynamics: A Nonequilibrium Green’s Function Approach, which unfortunately, is not valid because the expressions it gives do not give the correct result in the equilibrium limit. This is shown in

Quantum thermodynamics of the driven resonant level model 
 Anton Bruch, Mark Thomas, Silvia Viola Kusminskiy, Felix von Oppen, and Abraham Nitzan

What is striking to me about both papers is that they consider a non-interacting model, i.e. the Hamiltonian is quadratic in fermion operators and exactly soluble.
This shows just how far we are from any sort of theory of a realistic system, i.e. one with interactions and which is not integrable.

Phil Geissler gave a nice introduction to different theorems for fluctuations in the dissipation (defined as the difference between the entropy change and heat/temperature). The most general theorem is that due to Gavin Crooks and implies the Jarzynski inequality, the fluctuation theorem, and the second law of thermodynamics.
A key question is what sorts of non-equilibrium processes (protocols) minimise the dissipation and whether the distribution is Gaussian (it often is).
He then described near optimal protocols to invert the magnetisation in a two-dimensional Ising model.

Suri Vaikuntanathan talked about coupled (classical) master equation models for biomolecular networks that have mathematical similarities to an electronic Su-Schrieffer-Heeger model which is an one-dimensional example of a topological insulator.
The work is described  in a preprint with A. Murugan,  "Topologically protected modes in non-equilibrium stochastic systems".
This is potentially  important because it may provide  "a framework for how biochemical systems can use non equilibrium driving to achieve robust function."

David Limmer gave a nice talk which considered thermodynamics as a large deviation theory and how that can even have meaning out of equilibrium and there is a notion of an entropy, a "free energy" and a "temperature". His slides are here.
A key notion is to focus on ensembles of trajectories rather than a probability distribution function. There are two alternative computational strategies: transition path sampling and diffusion Monte Carlo (the cloning algorithm).
He considered several concrete examples, such as thermal conductivity in carbon nanotubes, and electrochemical processes at electrode-water interfaces.

20 key concepts in thermodynamics and condensed matter

Tomorrow I am giving a summary lecture for the end of an undergraduate course PHYS2020 Thermodynamics and Condensed Matter. I taught the second half of the course, which has featured in some earlier posts. Here are the slides where I attempt to summarise 20 key ideas/results/concepts in the course.

My approach to the key ideas in thermodynamics is heavily influenced by Hans Buchdahl, my ANU undergraduate lecturer (and honours thesis supervisor) and his (dense) Twenty Lectures on Thermodynamics.
A similar axiomatic macroscopic approach which starts with the second law has more recently been championed by Elliot Lieb and Jacob Yngvason, and described in a nice Physics Today article.

What is the chemical potential?

I used to find the concept of the chemical potential rather confusing.
Hence, it is not surprising that students struggle too.
I could say the mantra that "the chemical potential is the energy required to add an extra particle to the system" but how it then appeared in different thermodynamic identities and the Fermi-Dirac distribution always seemed a bit mysterious.

However, when I first taught statistical mechanics 15 years ago I used the great text by Daniel Schroeder. He has a very nice discussion that introduces the chemical potential. He considers the composite system shown below, where a moveable membrane connects two systems A and B. Energy and particles can be exchanged between A and B. The whole system is isolated by the environment and so the equilibrium state is the one which maximises the total entropy of whole system.

Mechanical equilibrium (i.e. the membrane does not move) occurs if the pressure of A equals the pressure of B.

Thermal equilibrium (i.e. there is no net exchange of energy between A and B) occurs if the temperature of A equals that of B. Thus, temperature is the thermodynamic state variable that tells us where two systems are in thermal equilibrium.

Diffusive equilibrium (i.e. there no net exchange of particles between A and B) occurs if the chemical potential of particles in A equals that in B, where the chemical potential is defined as

Starting with this one can then derive various useful relations such as those between the Gibbs free energy and the chemical potential (dG= mu dN and G=mu N).
Thus, the chemical potential is the thermodynamic state variable/function that tells us whether or not two systems are in diffusive equilibrium.

Doug Natelson also has a post about this topic. He mentions the American Journal of Physics article on the subject by Ralph Baierlein, drawing heavily from his textbook. However, I did not find that article very helpful, particularly as he mostly uses a microscopic approach, i.e. statistical mechanics. (Aside: the article does have some interesting history in it though).
I prefer to first  use a macroscopic thermodynamic approach before a microscopic one as, I discussed in my post, What is temperature?

Low temperature physics without nuclear weapons

Liquid 3He is amazing stuff. Below temperatures of a few hundred milliKelvin it forms a model (and the original inspiration for) Landau Fermi liquid. Furthermore, below about 1 mK it forms two different superfluid states, involving Cooper pairs in a spin triplet state. This is the model case for unconventional superconductivity.

Liquid 3He is actually of great practical use since it the crucial ingredient of dilution refrigerations that allow cooling from a few Kelvin to temperatures as low milliKelvin.
But where do labs get 3He from?
Well, it is a very useful by-product of nuclear weapons production.
Currently, the scientific community (which consumes only about 1% of the supply) is experience supply problems and dramatic price increases (a 15-fold increase between 2004 and 2010).
Why is this happening?
Thankfully, we are cutting back on nuclear weapons production!

One practical way to solve this problem is to develop alternative materials for ultra-low temperature refrigeration; one possibility is by adiabatic demagnetisation. Indeed, this is the method that was first developed in the 1930s using paramagnetic salts to achieve temperatures below about 0.3 K (and was the basis of the 1949 Nobel Prize in Chemistry) and is the basis for nice undergraduate problems in thermodynamics and statistical mechanics. Simply the entropy is a function of B/T (where B is the magnetic field and T the temperature). One cools the system down in a fixed magnetic field, then adiabatic isolates it and reduces the magnetic field slowly. In the last step the entropy must not change and so the temperature must decrease. (This is shown as the red horizontal arrow in the figure below). This is also known as the magnetocaloric effect. The problem is that most paramagnetic materials are insulators and one would prefer to have a metallic material that is a good thermal conductor and can be "machined".

I learnt some of this from an interesting paper (that I actually looked at in preparing an undergraduate thermodynamics lecture about Maxwell relations).

Large magnetocaloric effect and adiabatic demagnetization refrigeration with YbPt2Sn 
Dongjin Jang, Thomas Gruner, Alexander Steppke, Keisuke Mitsumoto, Christoph Geibel and Manuel Brando

The authors mention some basic unanswered science questions about why this material is a good candidate. Specifically, why is the Kondo temperature (associated with interaction of the magnetic moments of the Yb3+ ions with the conduction electrons) and the inter-ion magnetic interactions so low? This ensures that the spins act essentially like non-interacting spins (with a large entropy) down to less than 1 K.

A key figure is below, showing the entropy versus temperature at several different magnetic fields.

Do you really need to use Keldysh Green's functions?

Or are you cutting butter with a chainsaw? Or using a helicopter to cross the road?

Previously I posted that Green's functions are just a technique.
Here I have a new point. When using them one has to make a choice about how one treats the time variable in the complex plane. This is rather technical, subtle, and confusing. The choices available include imaginary time (Matsubara), retarded and advanced, Kadanoff-Baym, Keldysh, ....

Matsubara Green's functions can only describe equilibrium properties. They reflect a beautiful and profound connection between imaginary time and temperature. However, due to the very powerful fluctuation-dissipation relation (embodied in the Kubo formula), they can also be used to calculate transport properties in the linear response regime. They are also the easiest to use, both analytically and computationally. Although for some numerical methods such as Quantum Monte Carlo the analytic continuation to real frequencies can be a can of worms.

Kadanoff-Baym and Keldysh are constructed to allow description of non-equilibrium states. A beautiful and profound application of them is the derivation of Boltzmann-type transport equations. I think when Kadanoff and Baym did this it was a major conceptual advance that further buttressed Fermi liquid theory.
A nice accessible introduction to Keldysh is the review by Rammer and Smith. Nevertheless, keeping track of the contours and the relation between Keldysh, retarded and advanced components of the Greens functions can quickly become overwhelming.

There is no doubt that far from equilibrium, Keldysh is necessary and good. However, my concern is that sometimes people do a lot of formalism with Keldysh and then in the end they just do some linear response calculation that they could have just done with Matsubara. I conjecture this applies to some of the examples considered by Rammer and Smith.

In my Ph.D I used Keldysh to consider the non-linear interaction of zero sound with order parameter collective modes in superfluid 3He-B. Much later I realised that I could probably have done the same calculations with Matsubara.

Emergence and singular asymptotic expansions

Seth Olsen kindly lent me his copy of Chemistry, Quantum Mechanics, and Reductionism by Hans Primas, published in 1981. I has a Foreword by Paul Feyerabend! 

[Primas died last October and there will be a symposium in his honour later this year]
This is a book I had wanted to read for a while since I had seen it referenced in various philosophical contexts. Besides some deep philosophy he has lots of polemical statements about theoretical chemistry.

Wanting to find an electronic version I could copy choice quotes from led me to a more dense, broader, and more recent (1998) article Emergence in exact natural science.

Here I mention a few highlights.

emergence and theory reduction are related.

Theory reduction is the process where a more general theory, such as quantum mechanics or special relativity, "reduces" in a particular mathematical limit to a less general theory such as classical mechanics. This is a subtle philosophical problem that is arguably poorly understood both by scientists [who oversimplify or trivialise it] and philosophers [who sometimes overstate the problem]. The subtleties arise because the two different theories usually involve concepts that are "incommensurate" with one another.

the distinction inside/outside is not covered by the most fundamental context-independent natural laws (first principles of physics). 

Here Primas is stresses the sometimes "arbitrary" value judgements that are made in distinguishing a "system" and its "environment". This involves distinguishing "patterns" and invoking "symmetry breaking".  He introduces notions of topology to try and make such distinctions more rigorous. I found this too technical to appreciate.

 Many inter-theoretical relations can be mathematically described by asymptotic expansions. Singular asymptotic expansions are never uniformly convergent in the intrinsic topology of the basic theory. This nonuniformity is not a disaster but an indication that the limiting case represents a caricature, suppressing irrelevant details and enhancing contextually relevant features. The discontinuous change in the limit leads to a discontinuous change in the semantics and therewith to a description in a new language in terms of emergent properties. In the same sense as a photograph can never replace a brilliant caricature, an asymptotic description can – for the intended purpose – be more adequate than the exact description.

Michael Berry also has a 1994 article that takes a similar point of view.

The assertions  

“something consists of elementary systems”,   

“something can be decomposed into “elementary systems”, 

“something can be described in terms of “elementary systems”, 

are not equivalent. 

When a light wave passes an object, a typical discontinuity – called the shadow – can be observed. However, in Maxwell’s electrodynamics – the fundamental theory for the propagation of light – shadows do not exist. Maxwell’s electrodynamics is governed by partial differential equations which have only continuous solutions. The discontinuities associated with shadows appear only in geometric optics, the limiting case of vanishing wavelength l , l → 0 .

He discusses at length how the notion of "molecular structure" in chemistry is an emergent concept. This relates to the issue of quantum entanglement between electrons and nuclei.

In a quantum theoretical description the molecular shape emerges by abstracting from the actually existing Einstein–Podolsky–Rosen correlations between the electrons and the nuclei. Historically, the structure concept has been introduced into quantum chemistry by the so-called Born-Oppenheimer approximation. But this terminology is misleading since the main issue is not an approximation, but the breaking of a holistic symmetry. A more proper appreciation of the Born–Oppenheimer-description stresses its singular nature: it is an expansion about the singular point of infinite nuclear masses. An asymptotic expansion can be formulated in terms of the ratio e = (m/M)^1/4, where m is the mass of an electron and M is a mean nuclear mass of the molecular system. In the limiting case e = 0 the holistic correlations between nuclei and electrons are suppressed so the description of a molecule reduces to the description of the motion of electrons in the electric field of a classical nuclear framework. In this description the molecular structure is a property described by an emergent classical observable. The singular limiting case     e = 0 leads to a discontinuous change in the description and is the starting point for an asymptotic expansion in terms of the emergent property at higher levels of description. 

He then gives a another example that was new to me.

The transition from the more fundamental Lorentz-relativistic quantum mechanics to Galilei-relativistic quantum mechanics is governed by the contraction of the Lorentz group to the Galilei group – a highly singular limit. While the Lorentz group is semisimple, the Galilei group is not but has a more complicated mathematical structure. The emergent quantity associated with this contraction is the mass in the sense of a classical observable (which commutes with all other observables and can therefore be treated as a real parameter).

He also discusses how the concept of temperature is emergent, emphasising the centrality of the zeroth law of thermodynamics. 

Probing non-equilibrium dynamics in a quantum many-body system

This week at UQ there was a fascinating Quantum Science Seminar by Jorg Schmiedmayer, describing some beautiful ultra cold atom experiments.  Much of the talk is nicely discussed in a book chapter Does an isolated quantum system relax? [Answer is yes].
Some of the most recent results are in a Science paper, Experimental observation of a generalised Gibbs ensemble that appeared this month.

The experiments involve a one-dimensional Bose gas that can be described as a Luttinger liquid. This means that many properties, even non-equilibrium ones, can be calculated analytically and compared to experiment. This is a theorists dream!

Here are a few things that stood out.

Relaxation from a non-equilibrium state to the thermal equilibrium state occurs on several different time scales. First there is rapid relaxation to a "quasi-steady state" described by a Generalised Gibbs ensemble [This idea goes back to Jaynes] that involves an effective temperature [that one can even theoretically calculate in terms of the bose gas interaction strength].

There is a characteristic length scale associated with the relaxation.

One can even directly measure higher order correlation functions (4th, 6th and 10th order!), as seen below. Furthermore, in a Luttinger liquid [a non-interacting boson field theory] these should factorise in terms of the 2nd order correlation function [reflecting Wick's theorem]. One can even test this experimentally.

One can also simulate the sine Gordon theory and vary the coupling constant and so move through the associated phase transition.

What is temperature?

Doug Natelson, writes one of the few blogs on condensed matter physics. He recently wrote a blog on What is temperature? I don't really like his perspective. I think it is best to define temperature in macroscopic and operational terms, using the zeroth law of thermodynamics. The attached slides are from my undergraduate lectures. The key ideas are

* the zeroth law allows us to assign a single number to a thermodynamic system that has the important property that this number will tell us whether or not the system will change when it is brought into thermal contact with another system.

* a thermometer is just a thermodynamic system with just one state variable.