Derivation of the Heat Equation

Historical Preliminaries – Temperature and the First Law of Thermodynamics

The ancient Greek engineer Philo of Byzantium in 220 BCE noted that air expands when heated and contracts when cooled.

In 1612, building on the exploratory work of Galileo Galilei, the physician Santorio Santorio added a scale to a gas thermoscope, thus creating an instrument that quantitatively measures the change in density of gas with temperature. German physicist Daniel Gabriel Fahrenheit refined the concept, noting that liquid mercury expanded when heated and created the first reliable mercury-based thermometer in 1714.

In 1742 the Swedish astronomer Anders Celsius defined a temperature scale related to the boiling and freezing phases of water. He used 100C to be the temperature of ice, and 0C to be the temperature of boiling water. This convention was inverted in the modern era.

In studying the change in density of low-pressure gases, Joseph Louis Gay-Lussac in 1808 found that there was a univeral constant that related the volume ratio of any gas at one temperature relative to some fixed temperature. Using the modern Celsius scale the freezing point of water as the reference, the relation is $R(t) = \frac{V(t)}{V(0)} = \left(1 + \frac{t}{\alpha}\right)$ where $t$ is the temperature in Celsius and $\alpha = 273.15 [C]$ . This relation was valid over the range for which experimentation was possible. However, exrapolating outside this range, the zero-volume temperature implied by the experimental relation was $t = -\alpha$ . It is natural to define a new temperature scale $T = t + \alpha^{-1}$ the ratio is $S(T) = R(t) = T$ . When the gas was constrained to a linear tube with a smooth piston separating the evacuated portion from the gas-containing portion, a gas theromometer analouge was created.

Researchers tried to relate the effect of weight to temperature. By increasing mass acting on the piston, the gas was compressed and the density increased, and by decreasing the mass acting on the piston, the gas expanded. The degree of expansion and contraction was determined by the mass. By heating a compressed gas, the same volume could be attained as the uncompressed gas, cancelling the work done on the gas by the mass. The temperature of the gas could be determined by removing the mass. Hence the work done by the mass on the gas corresponded to an increase in the temperature of the gas. This relation was described by $c\Delta T + mg\Delta h = 0$ where $T$ is the temperature. Hence $cT + mgh = cT_0 + mgh_0 = E$ . Since $cT$ has the units of energy, the constant $c$ was called the heat capacity. Observations such as these culminated in the statement of the first law of thermodynamics, a result co-developed by many scientists in the mid 1840s, most notably the English physicist James Prescott Joule.

Conducting adiabatic experiments, the heat capacity of materials could be computed, and was found in general to be dependent on the material and a function of the temperature of the material.

Derivation of the Heat Equation

Consider a linear filament of some uniform composition and known density, fixed cross sectional area $A$ and heat capacity. Attach an axis with units to the the filament such that at position $x$

Subdivide the filament into small segments of width $dx$ , and let $q(x)$ be the heat flux across $A(x)$ in the direction established by the axis. Let $Q(x,t)$ be heat per unit mass generated by the material. Then

$c (\rho Adx) \partial_t T(x,t) = Aq(x,t)-Aq(x+dx,t) + Q(x,t)(\rho A dx)$

$c\rho \partial_t T(x,t) = -\partial_x q(x,t) + Q(x,t)$

It was proposed by the French mathematician and physicist Joseph Fourier in 1822 that heat flows in the opposite direction to the temperature gradient, which is the direction along which temperature increase is greatest. So

$q = -h \partial_x T$ for some constant $h$ called the thermal conductivity. Hence

$c\rho \partial_t T = h \partial_{xx} T + Q$

When the material does not produce its own heat, $Q = 0$ and the equation is

$\partial_t T = \alpha \partial_{xx} T$

which is the standard form of the homogeneous heat equation. The heat equation and its equivalents form a class of PDEs known as elliptical PDEs.

While the study of heat may seem relatively uninteresting, the study of heat and diffusion processes can be applied to financial mathematics, specifically the Black and Scholes equation, which is the a key modern tool in the pricing of derivative securities.

Derivation of the Heat Equation

Historical Preliminaries – Temperature and the First Law of Thermodynamics

Derivation of the Heat Equation

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