Category Archives: Einstein

Comments on Einstein Solid

November 22, 2017 – 8:47 AM

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Heat capacity of an Einstein solid as a function of temperature. Experimental value of 3Nk is recovered at high temperatures.
Reference: Disturbance Theory

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Einstein Solid – Wikipedia

The Einstein solid is a model of a solid based on two assumptions:

  • Each atom in the lattice is an independent 3D quantum harmonic oscillator
  • All atoms oscillate with the same frequency (contrast with the Debye model)

While the assumption that a solid has independent oscillations is very accurate, these oscillations are sound waves or phonons, collective modes involving many atoms. In the Einstein model, however, each atom oscillates independently. Einstein was aware that getting the frequency of the actual oscillations would be difficult, but he nevertheless proposed this theory because it was a particularly clear demonstration that quantum mechanics could solve the specific heat problem in classical mechanics.

A 3D quantum harmonic oscillator would be made up of high frequency, compacted cycles of electromagnetic field that have slowed down considerably due to their high inertia. Their motion is no longer linear, but a combination of circular, rotational and linear, which shows up as oscillatory.

Einstein treated each atom as an independent 3D harmonic oscillator, whose energy could only increase in quantum intervals of ‘hω’. Einstein assumed the same frequency for all atoms for the sake of simplicity.

The original theory proposed by Einstein in 1907 has great historical relevance. The heat capacity of solids as predicted by the empirical Dulong-Petit law was required by classical mechanics, the specific heat of solids should be independent of temperature. But experiments at low temperatures showed that the heat capacity changes, going to zero at absolute zero. As the temperature goes up, the specific heat goes up until it approaches the Dulong and Petit prediction at high temperature.

The classical mechanics predicts the heat capacity of solids to be independent of temperature. It did not explain the observed dependence at lower temperatures. Einstein could show this dependence with his quantum model even if not very accurately.

By employing Planck’s quantization assumption, Einstein’s theory accounted for the observed experimental trend for the first time. Together with the photoelectric effect, this became one of the most important pieces of evidence for the need of quantization. Einstein used the levels of the quantum mechanical oscillator many years before the advent of modern quantum mechanics.

The continuous change in properties, which is a feature of classical mechanics, is seen in context of normal dimensions. When we view properties at atomic dimensions, as in the case of black body radiation, the quantum effects of frequency cycles become prominent.

In Einstein’s model, the specific heat approaches zero exponentially fast at low temperatures. This is because all the oscillations have one common frequency. The correct behavior is found by quantizing the normal modes of the solid in the same way that Einstein suggested. Then the frequencies of the waves are not all the same, and the specific heat goes to zero as a T3 power law, which matches experiment. This modification is called the Debye Model, which appeared in 1912.

Einstein demonstrated the quantum effects on specific heats of solids at low temperatures which classical mechanics could not explain. Einstein did make simplified assumptions as regard the frequency of atoms, which were modified later in Debye model.

When Walther Nernst learned of Einstein’s 1906 paper on specific heat, he was so excited that he traveled all the way from Berlin to Zürich to meet with him.

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By vinaire | Comments (0)

Relativity: Math Imitating Life

December 3, 2011 – 1:36 PM

These are going to just some comments and references for a future article.

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Here is something by Albert Einstein to wonder about.

Relativity, The Special and General Theory

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In the first section of this book, Einstein makes the following points:

  1. Truth is relative to one’s experience. 
  2. If the experience is limited then one’s “truth” is limited.
  3. Euclidean Geometry is based on limited experience. 
  4. How do we consider a straight line? How do we consider distance?
  5. We assume locations in space to exist as if on a rigid body.

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Points from Section II:

  1. Let’s use a “rigid” Cartesian Coordinate System, and a rigid body as a unit, to describe positions in space.
  2. But, since optical observations are involved, let’s also take into account the properties of the propagation of light in determining the measurements.

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Points from Section III:

  1. Regarding “motion in space,” there is no such thing as an independently existing trajectory but only a trajectory relative to a particular body of reference. 
  2. The finiteness of the velocity of propagation of light would influence the perception of change in position with time.

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Points from Section IV:

  1. A system of co-ordinates of which the state of motion is such that the law of inertia holds relative to it is called a “Galileian system of co-ordinates.” 
  2. We cannot use a system of coordinates rigidly attached to earth, because, in that system of coordinates, stars would appear to be moving in a circle in violation of the law of inertia.
  3. We assume a Galileian systems of co-ordinates, which are rigid but not attached to earth. 

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Points from Section V:

  1. Principle of Relativity: Natural phenomena run their course according to exactly the same general laws with respect to all Galileian co-ordinate systems that are translating uniformly, and not rotating or accelerating, relative to each other.
  2. We may assume, as our body of reference, a Galileian coordinate system K0, in which natural laws are capable of being formulated in a particularly simple manner.
  3. We may then assume K0 to be “absolutely at rest,” and all other Galileian systems K “in motion.”
  4. The “motion” of Galileian systems K shall contribute to their diminished simplicity, or increased complexity, of the formulation of natural laws.

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Points from Section VI:

  1. The theorem of the addition of velocities employed in classical mechanics, as we shall see, does not hold in reality.

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Points from Section VII:

  1. The velocity of light of any wave-length is constant in vacuum. It does not depend on the motion of the body emitting the light.
  2. The theorem of the addition of velocities employed in classical mechanics does not hold for the velocity of light, which remains the same with respect to all bodies of reference.
  3. This result comes into conflict with the principle of relativity set forth in Section V, because it would appear that different laws of propagation of light must necessarily hold for different coordinate systems.
  4. It seems that the principle of relativity must be rejected because the theoretical investigations into electromagnetic phenomena, leads conclusively to the constancy of the velocity of light in vacuo.
  5. In reality there is not the least incompatibility between the principle of relativity and the law of propagation of light, as shown in the special theory of relativity, by an analysis of the physical conceptions of time and space.

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Points from Section VIII:

  1. “Time” should be defined as a measure in the immediate vicinity (in space) of the event.

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Points from Section IX:

  1. Events that are simultaneous with reference to one co-ordinate system are not necessarily simultaneous with reference to another co-ordinate system because of the finite velocity of light.
  2. Every reference-body (co-ordinate system) has its own particular time.

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Points from Section X:

  1. Owing to the consideration of simultaneity, the measure of the same distance from two different coordinate systems, which are in relative motion to each other, is not necessarily the same .

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There is no loss of energy in a pure standing wave. In other words, the energy would be conserved. We may then look at this physical universe as an example of pure standing wave, because energy is conserved in this universe.

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The most fundamental phenomenon in this universe seems to be a back-and-forth motion around a reference value. Call it a vibration that creates waves; but this phenomenon seems to underlie all other phenomena.

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Light seems to be tied to the vacuum of the space itself rather than to anything existing in that space. In other words, light does not seem to travel relative to anything in space. The velocity of light is the same regardless of the motion of the frame of reference that is used.

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The velocity of light is finite . Einstein used this fact to query the very basis of perception on cosmic scales.

At atomic scales, light seems to condense as standing waves instead of reflect. It seems to be the fractal iteration of this condensation that appears as electrons, protons, neutrons, etc. This makes perception at atomic scales questionable too. Our perception seems to be limited to the middle band.

The frequency and wave-length of light corresponds in such a way that the velocity of light always remains the same. That is the basis of confirming that the universe is expanding using the redshift of the Doppler effect. We apply the same argument to the shift in the pitch of the sound of whistle of a passing train. This brings up the following questions:

(1) The velocity of sound is constant with respect to a medium, such as air. Would we perceive the same shift in the pitch of sound if the air is not all displaced and moved around  by the passing train?

(2) The velocity of light is constant in space. Can we treat space as a “medium” made up of not the usual material, but dark matter perhaps!

(3) Does this dark matter get disturbed by passing light, similar to the way air gets disturbed by the passing train?

(4) How does this dark matter appear to behave at atomic dimensions?

Note: This seems to be looking at the old ether theory again.

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Counting goes up to infinity; then that collection of infinity may be regarded as “one.” For example, Infinity of fundamental particles may take the shape of an apple. We may then count apples. Not all apples will have the same number of fundamental particles, but each would be regarded as a unit apple.

A brief introduction to infinity

The counting to infinity may be repeated with this new “one.”  This procedure may continue without limit. This procedure may be reversed without limit also.

This raises the question, “Do we have a rigid relationship between the “numbers” used to account for the atomic phenomena and the numbers used to account for the cosmic phenomena?”

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The following is the basic assumption underlying Euclidean Geometry, which was pointed out by Einstein:

Euclidean geometry assumes that points, directions and distances behave as if they are associated with a rigid body. We are conditioned to think this way because the rigid body of earth provides our frame of reference.

But the question remains, “Does the “fabric” of space behaves like a rigid body, and if not, then do the laws of Euclidean Geometry still apply?”

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Here is a very interesting comment on mechanical brain and free will by Alan Turing (AMT/B/5 Image 5):

The Turing Digital Archive

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[To be continued…]

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By vinaire | Comments (117)