On classification of physical effects by the graph types of their mathematical models

Introduction

Physical effects are the way of converting one set of physical quantities into another set, or producing one set of physical quantities from the other one:

(PQ1out, … ,PQnout) = F(PQ1in, … ,PQmin),

where PQiin is the i-th input physical quantity and PQjout is the j-th output physical quantity.

For example, Hall effect converts electric current I in a conductor of width W with charge carrier density of D and magnetic field B across it into a voltage difference V across the conductor:

V = - IB/DWe,

where e is the elementary charge.

Here PQ1in = I, PQ2in = B, PQ3in = D, PQ4in = W, and PQ1out = V.

Physical effects can be classified in various ways:

1. They can be classified by physical quantities they convert. (For example, all effects that convert mechanical deformation into voltage, such as piezoelectric effect, under this classification have to be placed into one class.)
2. They can be classified by what can be accomplished with the help of the certain inputs.
For example, by applying a high pressure the following things can be accomplished:
• - non-conductors can be converted into conductors [1];
• - brittle materials can be converted into ductile ones [2];
• etc.
3. They can be classified by how to achieve a ceratin result.
For example, non-conductors can be converted into conductors by:
• - applying high pressure [1];
• - immersing them into the atmosphere of bromine vapor;
• etc.
4. They can be classified by the type of the dependence between inputs and outputs. (For example, all effects with a linear dependence between inputs and outputs constitute one class under such a classification.)

Classification 1), 2), and 3) are trivial and, in fact, were already partially accomplished by various encyclopedia. Classification of effects by the type of the dependence between inputs and outputs is less obvious and so far has been accomplished by no one. This article is an attempt to fill in this gap and present excerpts from the classification of the latter type.

1. Effects with continuous dependences between inputs and outputs:

• 1.1. Effects with continuous increases:

There are tons and tons of them. I will only list those, which are needed to contrast stepwise dependencies between the same inputs and outputs. Specifically:

• 1.1.1. Effects with continuously increasing dependence of object's volume on something else:
1. Thermal expansion/contraction
• 1.1.2. Effects with continuously increasing dependence of the rate of a chemical reaction on something else:
1. The rate of most chemical reactions increases with temperature. This is so-called Arrhenius law. But there are exceptions.
• 1.1.3. Effects with continuously increasing dependence of material's conductivity on something else:
1. The conductivity of carbon resistors increases with temperature.
• 1.2. Effects with continuous decreases:

• 1.2.1. Effects with continuously decreasing dependence of object's volume on something else:
1. None at this time. Who can pick up this challenge ? (Don't think that thermal contraction belongs here, as volume contracts only when temperature drops, not rises.)
• 1.2.2. Effects with continuously decreasing dependence of the rate of a chemical reaction on something else:
1. None at this time. Who can pick up this challenge ?
• 1.2.3. Effects with continuously decreasing dependence of material's conductivity on something else:
1. Conductivity of most materials decreases with the temperature.

2. Effects with stepwise dependences between inputs and outputs:

• 2.1. Effects with stepwise increases:

• 2.1.1. Effects with stepwise increasing dependences between the volume of an object and something else:
1. Phase transitions of most solids. When melted, the volume of objects increases stepwise without a temperature change.
• 2.1.2. Effects with stepwise increasing dependences between the rate of a chemical reaction and something else:
1. None is known so far. Who can pick up this challenge ?
• 2.1.3. Effects with stepwise increases in the conductivity of a material:
1. Certain non-conductive materials can be converted into conductive ones by applying a high pressure above a certain threshold [1].
2. Graphite is a poor conductor. But in the atmosphere of the bromine vapor above a certain concentration it turns into a good conductor.
• 2.1.4. Effects with stepwise changes in the brittleness and ductility of metals:
1. Under high pressure above a certain threshold brittle metals turn into ductile ones [2].
• 2.2. Effects with stepwise decreases:

• 2.2.1. Effects with stepwise decreasing dependences between the volume of an object and something else:
1. Phase transitions of ice, cast iron, silicon, germanium, bismuth, antimony, gallium: they lose in volume when melted.

(It would be interesting to discover an effect of a stepwise volume change without a phase transition, though. Who can pick up this challenge?)

• 2.2.2. Effects with stepwise decreasing dependences bewtween the rate of a chemical reaction and something else:
1. Stepwise drop to 0 with the temperature: tungsten reacts with halogens only when temperature is less than 1200 degrees Celsius producing halides: W + 2I → WI2. It does not react when temperature is above 1200 degrees. Conversely, under temperatures higher than 1200 degrees halides dissociate back into tungsten and halogens: This is basis of the so called Halogen cycle employed in halogen lamps.
• 2.2.3 Effects with stepwise decreases in the conductivity of a material:
1. None is known at the present time. Who can pick up this challenge ?

3. Effects that convert a continuous input into a discrete output:

1. The stress-strain curve for most materials is a continuous line. But for small thin specimens of certain materials it is a stepwise line [3].

4. Effects that convert a discrete input into a continuous output:

• no examples at this time.

5. Effects that convert a monotonic input into an oscillating output

1. The degree of the partial reflection of light from a thin glass block periodically depends on the block's thickness [4].

6. Effects that convert an oscillating input into a monotonic output:

• no examples at this time.

7. Effects that invert the normal dependence between input and output

1. Soft substances are usually strengthened by adding hard metals. But cast iron is strengthened by adding soft metals, such as tin, zink, and lead [5].
2. Resistance of most conductors increases as the temperature rises. But resitance of carbon resitors, conversely, decreases as the temperature rises.

R E F E R E N C E S:

1. USSR invention certificate #833346
2. Percy W. Bridgman, Nobel Prize Lecture.
3. Properties of engineering materials, by Raymond A. Higgins, 2nd ed., Industrial Press Inc., 1994, NY (ISBN 0-8311-3055-5), p. 98.
4. QED: The Strange Theory of Light and Matter, by Richard Feynman, Princeton University Press, 1985 (ISBN 0-691-08388-6)
5. USSR invention certificate #644863