Exponential decay

For an exponential process is a process in which a size changes exponentially. A distinction is made between

• Exponential growth in the size grows faster and faster,
• Exponential decay with a size more and more slowly approaches zero, and
• Exponential approach, in which a size approaches a predefined value.

Here, the exponential decay is a special case of the exponential approximation to the predetermined value is zero.

Exponential Growth

If a process of growth in size, the growth rate (ie, the positive temporal change of variable) is proportional to the size itself is exponential growth before:

The proportionality constant is obtained from this proportionality, the differential equation

Whose solution is an exponential function:

This gives the meaning of a time period in which the size increases in each of the e -fold. is the value of the variable at the beginning ( at the time ).

Exponential decay

If the acceptance of a size proportionate to the value of the variable itself, it is called exponential decay, exponential decrease or exponential decay.

Examples

Time- exponential decay:

• Radioactive decay: Every second decomposes a fixed percentage of the existing atomic nuclei of the substance; present the fewer cores yet, the slower decreases their number
• Discharging a capacitor through a resistor
• Self-induction voltage when the voltage changes to a coil
• Current at turn-off of a coil
• Vibration amplitude of a damped pendulum ( in Stokes friction)
• Emptying a water tank through a thin tube on the ground: The deeper the water level falls, the lower the water pressure in the hose and the slower the water flows out
• Catalytic decomposition of substances by a chemical reaction, see exponential #, kinetics (chemistry) # first-order reactions.
• Relaxation ( NMR): reconstruction of the longitudinal or transverse decay of the nuclear magnetization after a fault.

Spatially exponential ( with the penetration depth ) decrease in:

• Absorption of some radiation in homogeneous material

Mathematical representation

Since the decline is a negative change, is the differential equation ( here written for temporal decrease ) now

And their solution is

Is therefore the time period in which the size of each drop on the times ( about 37 %). This is called time constant in physics and life.

A more descriptive size instead of the half-life. It specifies the time period within which the size always decreases by half, and can be easily calculated from the time constant:

Exponential convergence

In many physical processes, a physical size between two interconnected bodies / systems balances.

Examples:

• The temperature of a metal pad resembles itself to the ambient temperature.
• The temperatures of two different hot, thermally conductive metal blocks connected to each other at the same.
• The voltage of a capacitor to be charged at the charging voltage approaches.
• The current during the switching of a coil approaches to the given by Ohm's law current.
• The water levels of two differently filled, connected to a thin tube water tank are similar to each other.
• Diffusion: The concentrations of a substance in two interconnected chambers balance each other out.
• The rate of fall of a body in a fluid of finite viscosity approaches its terminal velocity at ( Stokes friction).

Many of these examples have in common is that two variables are related, an intensive quantity and extensive quantity:

• Temperature and amount of heat (thermal energy )
• Electric voltage and electric charge on the capacitor
• Water pressure and volume ( or mole ) in cylindrical containers
• Concentration and amount of substance

The two variables are proportional to each other, respectively, and a difference in the first size will cause a flow (or current ) of the second size flows between the two systems. This in turn causes a change in the systems of the first variable:

• A temperature difference causes a heat flow and thus temperature changes in both blocks.
• A voltage difference across the capacitor causes a flow of charge carriers and hence a change in voltage.
• A concentration gradient causes a particle flux and hence concentration changes.
• A Füllhöhendifferenz (→ pressure difference ) causes a flow of matter and thus Füllhöhenänderungen.

The change over time of the intensive size is proportional to the strength of the respective flow, and this is proportional to the difference in size. In such a case, therefore, applies to a size of the differential equation

This fundamental issue is for the phenomena described above same, therefore knowledge and laws between these can also be easily transferred. The laws of diffusion, for example, apply equally to the conduction of heat and electrical charge. ( However, Electrical phenomena are usually very fast. For liquids / gases without excessive friction / damping, the inertia of the moving mass for additional effects, makes, usually in the form of vibrations and sound waves. )

If one of the two values ​​is constant ( temperature, charging voltage ), then the size considered is closer to this value. If both values ​​are variable, so they will approach each other. In both cases, the values ​​approach to a final value, which can be most easily calculated.

As a differential equation can be written

With the solution

This is the value of at the beginning ( at the time ).

The exponential drop is as an approximation to the value 0 is a special case of the exponential approximation with.

The final value AEnd is never actually achieved, but only getting better approximated; In practice, the smaller and smaller difference to the final value will eventually be smaller than usual measurement inaccuracies. After five times the time constant (), the original difference has already dropped to less than 1% after seven-fold () to less than 1 ‰.

The time constant can be determined in a specific case and depends on variables such as general resistances and capacitances:

• For the charging or discharging of a capacitor to the electric capacitance through a resistor to the electric resistance can be simply replaced in the above equation by the voltage, and then straight.