The temperature a molecule does not have

Equipartition says half a kT for every quadratic way of storing energy, and it is one of the most useful statements in the subject. What it says, exactly, is that the average is half a kT. It says nothing about how far from that average a system is at any moment, and for a large system the omission does not matter because the answer is “not far at all”.

For a small system it matters entirely.

The narrow curve is a system of ten thousand degrees of freedom and its energy is definite to a per cent and a half. The broad one is six — a single small molecule, with three ways of translating and three of rotating — and its energy at any instant is anywhere within a factor of two of the average.

Where the width comes from #

The result follows from the Boltzmann factor in two lines and is worth doing because it produces a relation between two quite different kinds of quantity.

The probability of a system having energy $E$ is proportional to $\Omega(E)\,e^{-E/kT}$ — the number of states at that energy times the exponential that decides everything. The first factor rises steeply with energy and the second falls steeply, so the product is sharply peaked; expanding the logarithm of the product about its maximum gives a Gaussian whose variance is

$$\langle (\Delta E)^2\rangle = kT^2 C.$$

On the left is a fluctuation — how much a quantity wanders when nothing is being done to it. On the right is a response — how much the energy changes when the temperature is changed. They are the same number, and there is no reason from either side alone to expect that they would be.

That pairing is the first instance of a pattern that runs through the whole subject. The jiggling of a suspended particle is tied to the viscosity that damps it; the voltage noise across a resistor is tied to its resistance; the fluctuation of a magnetisation is tied to its susceptibility. In each case a quantity that describes a system’s restlessness at equilibrium is equal to a quantity describing how it answers a push, and the reason is always the one above — both are derivatives of the same logarithm.

How large a thing has to be #

Because the heat capacity is proportional to the number of degrees of freedom and the mean energy is too, the relative fluctuation goes as one over the square root of that number. That single exponent decides where thermodynamics applies.

At a mole, the relative fluctuation is $10^{-12}$. Nothing measures that, so the energy of a macroscopic sample at a fixed temperature is a definite number and the whole apparatus of classical thermodynamics — with its exact differentials and its state functions — is exactly right.

At $10^{5}$ degrees of freedom, which is a particle ten nanometres across, it is about half a per cent. That is measurable, and it is measured: the fluctuating force on a small object in a fluid, the flicker in the light scattered from a colloid, the noise in a nanoscale calorimeter. At this size thermodynamics still applies and its fluctuations have to be carried along beside it.

At six, it is 58 per cent. Here the language breaks down. Saying that a molecule “is at 300 kelvin” is a statement about what it is in contact with, not a property it possesses — and asking for its energy at an instant is asking for a draw from a broad distribution.

The transition is gradual and the exponent is what makes it feel abrupt. A factor of a hundred in size is only a factor of ten in the fluctuation, so the region where the answer is ambiguous stretches over several decades, and where the line is drawn depends on what is being asked.

The straight line is worth pausing on because it is the experimental foundation of everything above.

The distributions drawn in the first figure are theory. Their tails — the rare high-energy states — are what a rate measurement samples, and a rate that follows an exponential in one over the temperature over many decades is a direct measurement that the tail falls off as the Boltzmann factor says it does. Reaction rates have been shown to do this over ten or more decades in rate, which is a far more stringent test than any measurement of a mean.

That is a general point about testing statistical mechanics. A mean is insensitive to the shape of a distribution and a threshold is not. Anything with a barrier in it — a chemical reaction, a nucleation event, an escape from a trap — samples the tail, and measuring how the rate depends on temperature measures the shape of a part of the distribution that no direct measurement can reach.

What the fluctuation is worth measuring for #

The relation runs both ways, and the reverse direction is a technique.

If the mean square energy fluctuation is $kT^2C$, then measuring how much a small system’s energy wanders measures its heat capacity — without adding any heat to it, and without knowing anything about what it is made of. That is worth having whenever a sample is too small to warm measurably: a nanoscale calorimeter can be operated as a thermometer watching itself, and its own fluctuations report the quantity a calorimeter is built to find.

The same idea, applied to the position of a trapped particle rather than to an energy, is the standard calibration of an optical trap. The trap is a spring; the particle’s mean square displacement is $kT$ divided by the stiffness; so watching a bead jiggle in the trap, with no force applied and nothing known about the laser, gives the stiffness directly. Every measurement of a piconewton force in a biological experiment rests on it.

And in reverse again: knowing the heat capacity puts a floor on how quiet a small system can be. The temperature resolution of a bolometer is limited by its own energy fluctuations, so making it colder and smaller improves the sensitivity until the fluctuation floor is reached and then does not. The relation is a design constraint before it is a measurement.

The connection between the two figures is the one that makes the whole subject matter to chemistry and to biology.

A reaction with a barrier of half an electronvolt is essentially never crossed by a molecule with the average thermal energy: half an electronvolt is twenty kT at room temperature, and the mean molecule is nowhere near it. What crosses the barrier is the tail — the rare molecules that happen, at that moment, to hold twenty times the average. The rate is set entirely by how often that happens, which is what the Arrhenius exponential measures.

So the fluctuation is not a nuisance riding on top of a mean. It is the thing that does the work. A world in which every molecule held exactly its average energy would have no chemistry at all: nothing would ever cross a barrier, and every reaction with an activation energy above a few kT would simply not proceed.

That inverts the usual reading of thermodynamic averages. The average tells what a system is doing; the fluctuation tells what it is capable of, and for anything with a threshold in it the second is the interesting one.

When “temperature” stops meaning anything #

It is worth being precise about which statement fails first, because two things are usually conflated.

The energy of a small system is not definite, and that is what the figures show. But its temperature is a different quantity: it is a property of the reservoir the system is in contact with, and the reservoir is by assumption large. So a molecule in a gas has a perfectly definite temperature — the gas’s — and a completely indefinite energy, and no contradiction arises.

What breaks down is the case where there is no reservoir. An isolated system of a few particles has an energy and no temperature at all: temperature is defined by the rate at which the number of states grows with energy, and for a handful of particles that derivative is a jagged quantity rather than a smooth one. Asking for the temperature of five atoms in a box is asking for the slope of a staircase.

The practical version of the distinction is worth carrying. Temperature is a property of contact, not of matter. Anything in equilibrium with a large bath has the bath’s temperature, however small it is; anything not in contact with a bath has a temperature only to the extent that it is large enough for its own state counting to be smooth.

The one that made the sky blue #

There is a fluctuation whose consequences are visible without any instrument, and it belongs here rather than in optics.

A gas at equilibrium does not have a uniform density. The number of molecules in any small region fluctuates, by the square root of the number in it, and a region with more molecules has a slightly higher refractive index. Light passing through therefore meets a medium that is not optically uniform, and it scatters — from the fluctuations rather than from the molecules, which is the correct modern statement of why the sky is blue.

The two accounts agree numerically, and the reason is the one above: for an ideal gas the fluctuation in the number is exactly the number, so scattering from independent molecules and scattering from density fluctuations give the same answer. They stop agreeing where the fluctuations stop being independent — near a critical point, where the compressibility diverges and so does the density fluctuation, and a fluid that was clear becomes milky. That is critical opalescence, and it is the same relation between a fluctuation and a response with the compressibility in place of the heat capacity.

How small a thermometer can be #

The size question has a practical form: what is the smallest thermometer that can report a temperature to a stated precision?

A thermometer works by coming to equilibrium with what it is measuring and then having its own energy read. Its energy fluctuates by $\sqrt{kT^2C}$, so the temperature inferred from a single reading is uncertain by $\sqrt{kT^2C}/C = T\sqrt{k/C}$ — a resolution that depends only on the heat capacity, and is a millikelvin at room temperature for a capacity of about a millionth of a joule per kelvin.

Two things follow that are not obvious. Averaging helps, but only as the square root of the number of independent readings, and readings separated by less than the thermometer’s own relaxation time are not independent — so the useful figure is a resolution per root hertz rather than a resolution.

And making the thermometer smaller always makes it worse in this respect while making it better in every other. A smaller sensor responds faster, disturbs less and localises better, and it has a larger fluctuation floor. Every design of a small calorimeter is a choice on that curve, and the curve is the one drawn above with the axes relabelled.

Where the model stops #

The Gaussian shape is an expansion. The distributions drawn here are exact for a system of independent quadratic degrees of freedom and become Gaussian for large numbers by the ordinary central-limit argument. For a small system with a structured spectrum — a two-level system, a molecule with a few widely spaced vibrational levels — the distribution is not smooth at all and the variance is a poor summary of it.

The heat capacity is assumed constant over the width of the fluctuation. Where it is not, the relation acquires corrections, and near a phase transition it fails badly because the capacity is changing on the scale the energy wanders over. That is not a small technicality: it is the regime where fluctuations dominate and where the whole apparatus of critical phenomena begins.

The system is assumed to be in equilibrium with something large. The canonical distribution used throughout is the distribution of a system in contact with a bath at fixed temperature, and it says nothing about a system that is driven, isolated, or coupled to something comparable in size.

And the counting of degrees of freedom is classical. A degree of freedom that is frozen out contributes nothing to the capacity and therefore nothing to the fluctuation, so the effective number at room temperature is smaller than the count of coordinates. The exponent survives; the number on the axis does not.

The oldest measurement of Avogadro’s number #

There is a historical instance worth ending on, because it turned the fluctuation relation into a number that mattered.

The relative size of a fluctuation goes as one over the square root of the number of particles, so measuring a fluctuation counts the particles. Einstein saw this in 1905 and applied it to a suspended grain: the mean square displacement of a particle in water after a time t is proportional to that time, and the constant of proportionality contains the gas constant divided by Avogadro’s number. Perrin measured the displacements under a microscope between 1908 and 1911 and got a value good to a few per cent.

The importance was not the precision, which was ordinary. It was that the same number came out of five completely different measurements — the Brownian displacement, the rotational jiggling, the sedimentation profile, the critical opalescence of a fluid, and the blue of the sky — all of them fluctuation measurements, and all of them agreeing. That was what settled whether atoms were real, and it is the argument that the second experiment cannot disagree.

The reason the argument works is the one this essay is about: a fluctuation is a measurement of how many things there are, because a mean is not. Averages hide the granularity and fluctuations report it, and a set of measurements that agree on the granularity is agreeing on something a mean-field theory has no room for at all.

What the pictures cannot show #

The distributions draw a probability against energy and cannot show a timescale. How long a system takes to wander from one side of its distribution to the other is a separate question, decided by how strongly it is coupled to its bath, and nothing in equilibrium statistical mechanics answers it. A nanoparticle whose energy fluctuates by a per cent might do so a million times a second or once an hour, and which it is decides whether a measurement sees an average or a fluctuation.

The scaling figure draws a line through nine orders of magnitude and hides that the physics along it is not the same. At the small end the degrees of freedom are quantised and countable; in the middle they are classical and numerous; at the large end the system has phase transitions, spatial structure and its own internal correlations. The relation survives all three, which is remarkable and is not visible in a straight line.

Where the ladder goes next #

The equipartition ladder began with half a kT for every way of moving and continued with the share that is not half a kT. This rung asks how definite the share is. The rungs after it: the fluctuation–dissipation theorem in its general form, where the response to a small force at any frequency is fixed by the spontaneous fluctuations at that frequency; the noise floor of a measurement, where the same relation sets a limit no design removes; and the fluctuations of quantities other than energy — density, magnetisation, number — each tied to its own susceptibility by the same argument.

The habit worth carrying away is to ask what a thermodynamic quantity is an average of. Every one of them is, and the width of the thing being averaged is a physical quantity in its own right — usually equal to a response coefficient, usually negligible for anything large, and usually the whole subject for anything small.