Thermodynamic fluctuations and statistical physics
Posted Juli 26, 2009on:
Thermodynamic fluctuations and statistical physics
Main article: statistical physics
Einstein’s earliest papers were concerned with thermodynamics. He wrote a paper establishing a thermodynamic identity in 1902, and a few other papers which attempted to interpret phenomena from a statistical atomic point of view.
His research in 1903 and 1904 was mainly concerned with the effect of finite atomic size on diffusion phenomena. As in Maxwell’s work, the finite nonzero size of atoms leads to effects which can be observed. This research, and the thermodynamic identity, were well within the mainstream of physics in his time. They would eventually form the content of his PhD thesis.
His first major result in this field was the theory of thermodynamic fluctuations. When in equilibrium, a system has a maximum entropy. According to the statistical interpretation, the entropy can fluctuate a little bit. Einstein pointed out that the statistical fluctuations of a macroscopic object, like a mirror suspended on spring, would be completely determined by the second derivative of the entropy with respect to the position of the mirror. This makes a connection between microscopic and macroscopic objects.
Searching for ways to test this relation, his great breakthrough came in 1905. The theory of fluctuations, he realized, would have a visible effect for an object which could move around freely. Such an object would have a velocity which is random, and would move around randomly, just like an individual atom. The average kinetic energy of the object would be kT, and the time decay of the fluctuations would be entirely determined by the law of friction.
The law of friction for a small ball in a viscous fluid like water was discovered by George Stokes. He showed that for small velocities, the friction force would be proportional to the velocity, and to the radius of the particle (see Stokes’ law). This relation could be used to calculate how far a small ball in water would travel due to its random thermal motion, and Einstein noted that such a ball, of size about a micron, would travel about a few microns per second. This motion could be easily observed with a microscope. Such a motion had already been observed with a microscope by a Botanist named Brown, and had been called Brownian motion. Einstein was able to identify this motion with the motion predicted by his theory. Since the fluctuations which give rise to Brownian motion are just the same as the fluctuations of the velocities of atoms, measuring the precise amount of Brownian motion using Einstein’s theory would show that Boltzmann’s constant is nonzero. It would measure Avogadro’s number.
These experiments were carried out a few years later, and gave a rough estimate of Avogadro’s number consistent with the more accurate estimates due to Max Planck’s theory of blackbody light, and Robert Millikan’s measurement of the charge of the electron. Unlike the other methods, Einstein’s required very few theoretical assumptions or new physics, since it was directly measuring atomic motion on visible grains.
Einstein’s theory of Brownian motion was the first paper in the field of statistical physics. It established that thermodynamic fluctuations were related to dissipation. This was shown by Einstein to be true for time-independent fluctuations, but in the Brownian motion paper he showed that dynamical relaxation rates calculated from classical mechanics could be used as statistical relaxation rates to derive dynamical diffusion laws. These relations are known as Einstein relations.
The theory of Brownian motion was the least revolutionary of Einstein’s Annus mirabilis papers, but it had an important role in securing the acceptance of the atomic theory by physicists.
Main article: History of special relativity
His 1905 paper on the electrodynamics of moving bodies introduced the radical theory of special relativity, which showed that the observed independence of the speed of light on the observer’s state of motion required fundamental changes to the notion of simultaneity. Consequences of this include the time-space frame of a moving body slowing down and contracting (in the direction of motion) relative to the frame of the observer. This paper also argued that the idea of a luminiferous aether—one of the leading theoretical entities in physics at the time—was superfluous. In his paper on mass–energy equivalence, which had previously considered to be distinct concepts, Einstein deduced from his equations of special relativity what has been called the twentieth century’s best-known equation: E = mc2. This equation suggests that tiny amounts of mass could be converted into huge amounts of energy and presaged the development of nuclear power. Einstein’s 1905 work on relativity remained controversial for many years, but was accepted by leading physicists, starting with Max Planck.
Main article: Photon
In a 1905 paper, Einstein postulated that light itself consists of localized particles (quanta). Einstein’s light quanta were nearly universally rejected by all physicists, including Max Planck and Niels Bohr. This idea only became universally accepted in 1919, with Robert Millikan’s detailed experiments on the photoelectric effect, and with the measurement of Compton scattering.
Einstein’s paper on the light particles was almost entirely motivated by thermodynamic considerations. He was not at all motivated by the detailed experiments on the photoelectric effect, which did not confirm his theory until fifteen years later. Einstein considers the entropy of light at temperature T, and decomposes it into a low-frequency part and a high-frequency part. The high-frequency part, where the light is described by Wien’s law, has an entropy which looks exactly the same as the entropy of a gas of classical particles.
Since the entropy is the logarithm of the number of possible states, Einstein concludes that the number of states of short wavelength light waves in a box with volume V is equal to the number of states of a group of localizable particles in the same box. Since unlike others, he was comfortable of the statistical interpretation, he confidently postulates that the light itself is made up out of localized particles, since this is the only reasonable interpretation of the entropy.
This leads him to conclude that each wave of frequency f is associated with a collection of photons with energy hf each, where h is Planck’s constant. He does not say much more, because he is not sure how the particles are related to the wave. But he does suggest that this idea would explain certain experimental results, notably the photoelectric effect.
Quantized atomic vibrations
Main article: Einstein solid
Einstein continued his work on quantum mechanics in 1906, by explaining the specific heat anomaly in solids. This was the first application of quantum theory to a mechanical system. Since Planck’s distribution for light oscillators had no problem with infinite specific heats, the same idea could be applied to solids to fix the specific heat problem there. Einstein showed in a simple model that the hypothesis that solid motion is quantized explains why the specific heat of a solid goes to zero at zero temperature.
Einstein’s model treats each atom as connected to a single spring. Instead of connecting all the atoms to each other, which leads to standing waves with all sorts of different frequencies, Einstein imagined that each atom was attached to a fixed point in space by a spring. This is not physically correct, but it still predicts that the specific heat is 3NkB, since the number of independent oscillations stays the same.
Einstein then assumes that the motion in this model are quantized, according to the Planck law, so that each independent spring motion has energy which is an integer multiple of hf, where f is the frequency of oscillation. With this assumption, he applied Boltzmann’s statistical method to calculate the average energy of the spring. The result was the same as the one that Planck had derived for light: for temperatures where kBT is much smaller than hf, the motion is frozen, and the specific heat goes to zero.
So Einstein concluded that quantum mechanics would solve the main problem of classical physics, the specific heat anomaly. The particles of sound implied by this formulation are now called phonons. Because all of Einstein’s springs have the same stiffness, they all freeze out at the same temperature, and this leads to a prediction that the specific heat should go to zero exponentially fast when the temperature is low. The solution to this problem is to solve for the independent normal modes individually, and to quantize those. Then each normal mode has a different frequency, and long wavelength vibration modes freeze out at colder temperatures than short wavelength ones. This was done by Debye, and after this modification, Einstein’s quantization method reproduced quantitatively the behavior of the specific heats of solids at low temperatures.
This work was the foundation of condensed matter physics.
Adiabatic principle and action-angle variables
Main article: Old quantum theory
Throughout the 1910s, quantum mechanics expanded in scope to cover many different systems. After Ernest Rutherford discovered the nucleus and proposed that electrons orbit like planets, Niels Bohr was able to show that the same quantum mechanical postulates introduced by Planck and developed by Einstein would explain the discrete motion of electrons in atoms, and the periodic table of the elements.
Einstein contributed to these developments by linking them with the 1898 arguments Wilhelm Wien had made. Wien had shown that the hypothesis of adiabatic invariance of a thermal equilibrium state allows all the blackbody curves at different temperature to be derived from one another by a simple shifting process. Einstein noted in 1911 that the same adiabatic principle shows that the quantity which is quantized in any mechanical motion must be an adiabatic invariant. Arnold Sommerfeld identified this adiabatic invariant as the action variable of classical mechanics. The law that the action variable is quantized was the basic principle of the quantum theory as it was known between 1900 and 1925.
Main article: Wave-particle duality
Although the patent office promoted Einstein to Technical Examiner Second Class in 1906, he had not given up on academia. In 1908, he became a privatdozent at the University of Bern. In “über die Entwicklung unserer Anschauungen über das Wesen und die Konstitution der Strahlung” (“The Development of Our Views on the Composition and Essence of Radiation”), on the quantization of light, and in an earlier 1909 paper, Einstein showed that Max Planck’s energy quanta must have well-defined momenta and act in some respects as independent, point-like particles. This paper introduced the photon concept (although the name photon was introduced later by Gilbert N. Lewis in 1926) and inspired the notion of wave–particle duality in quantum mechanics.