# Realistically, how can you double 20 quickly

## Exponential growth - explained in a way that is understandable for laypeople

Extra Articles> Exponential Growth

Author: Dr. Rüdiger Paschotta

The term exponential growth comes across in a wide variety of contexts - currently in discussions of the coronavirus-triggered Covid-19 pandemic. Here, assessments often fail because the characteristics of exponential growth are not fully understood. This problem occurs again and again in other contexts. Therefore, an attempt should be made here to explain this matter comprehensively to laypeople. It deals with the following central questions:

• What exactly does exponential growth mean?
• What are its special characteristics?
• When does exponential growth occur and why is it seen so often?
• How can you graph this type of growth?
• Can exponential growth go on indefinitely?
• How do you understand exponential decay and where does it occur?

Only an absolute minimum of mathematical formulas should be used here. Even if you lack the necessary basic knowledge for this, the majority of the article should still be understandable; only you cannot check some of the findings yourself.

We will discuss the basic principles using a number of interesting examples - in a wide variety of topics such as bacterial growth, epidemics, capital investment, viral messages, atom bombs, nuclear reactors, laser technology and oscillators. So it should be quite an educational thing.

### What exactly is exponential growth?

Simple answer to a simple question: This is a growth that is caused by a so-called Exponential function can be described. For example, the development of a variable over time A. - for example the number of bacteria in a nutrient solution - with exponential growth can be described by the following formula with the natural exponential function:

If you are having a little trouble doing the math here, don't worry - you will still be able to understand most of this article.

often also as A.(t) = A.0 Exp (αt) is written. It says t for time, and A.(t) means the value of A. for now t. For now t = 0, which is often chosen for the beginning of the consideration, we get the initial value: A.(0) = A.0because an exponential function always results in 1 when applied to the value zero. (Exception: 00 is undefined.) The symbol e stands for the so-called Euler's number and is approximately 2.71828. (Other exponential functions use a different “basis” than e, e.g. B. often 2 or 10.) In addition, α is a constant, i. H. a value that does not change over time; they are often referred to as Growth coefficient.

Of course, the size under consideration grows A.(t) the faster, the larger α is. With a calculator, you can eαt for any values ​​of α and t calculate easily; the key for using the exponential function is usually labeled EXP.

How am I supposed to multiply an odd number of factors?

Powers with an integer exponent are encountered relatively early in school. For example, exp (5) =e5 = e e e e e (5 factors e). How to do this for an "odd" number of factors such as B. 2.478 should perform, remains a mystery at first. You already guessed: There must be a completely different method of evaluating the exponential function that works for any, not just integer values; this discussion would go too far here.

You should not confuse exponential functions with power functions. The difference is that with an exponential function we have a fixed base (e.g. e or 10), while the exponent is variable; it is the other way around for power functions. Incidentally, an exponential function increases faster than in the long run each Power function, no matter how high its exponent is.

All of this is first and foremost a mathematical thing. What has not yet been explained: Why does something like this happen often in reality, or under what circumstances? Before we get into that, however, we should study an important property of the exponential function:

### The speed of growth

First I would like to explain what one understands by a growth rate: It is the increase (or general change) in the respective size per unit of time. Mathematically: the change in size (referred to as there) divided by the length of the time interval: there/d t.

In the example of bacterial growth, where the size of interest A. is the current number of bacteria is the rate of growth (referred to in a formula as there/d t or A.'(t)) the increase in this number per second, hour or day - depending on which time unit we use for it.

When the growth rate is constant, the size increases smoothly (linearly), but in general the growth rate changes over time.

How do you get the growth rate for a moment instead of a time interval?

A mathematical detail: So far, the growth rate would not be defined for a specific point in time, but for a more or less large time interval (e.g. one hour). But mathematically you can determine the growth rate for a certain moment by determining the so-called limit value for ever shorter time intervals. This is known as temporal Derivation - a central concept of the so-called Differential calculus. Strictly speaking, this limit value formation does not work precisely for the example of bacteria, but such details are not of central importance for the following considerations.

Especially for exponential Growth can be shown to have the following very simple relationship:

In the case of exponential growth, the growth rate is equal to the growth coefficient (a constant value) multiplied by the current value of the size. In a nutshell: The rate of growth is proportional to the size itself. It doesn't get any easier than that. For example, if A. doubled after some time, the rate of growth also doubled.

The consequence of this is that growth accelerates more and more. We'll look at that in more detail later.

### When does exponential growth occur?

When growth is exponential, the rate of growth is proportional to the value of the growing size itself.

Now it will hardly surprise you that the reverse is also true:

When the rate of growth is proportional to size itself, growth is exponential.

And the latter condition is quite often fulfilled in reality, at least approximately. Two examples of this:

• When the number of bacteria in a nutrient solution has increased tenfold, the number of new bacteria formed per second by cell division also increases tenfold, because ten times more bacteria can divide. (Of course, this does not always have to be the case; we will discuss this later.)
• If your savings in a savings account have doubled, you will also receive twice as much interest every year, which you can invest again.

It is often the case that growth is precisely determined by what's already there, and that is precisely why we encounter exponential growth so often in reality, at least approximately.

Physicists and engineers are constantly working with such differential equations.

In physics and technology, the development of a variable over time is often described by a differential equation - an equation that establishes a relationship between a variable, its derivative and possibly other things. If it has the mentioned simple form - a derivative proportional to the instantaneous value of the quantity - one recognizes this immediately and can give an exponential function as the solution. Solution here means a function that fulfills the differential equation and, if necessary, additional conditions.

### Example: bacteria in a nutrient solution

Let us assume that initially (i.e. currently t = 0) have 1000 bacteria in a nutrient solution in a test tube (i.e. A.0 = 1000) and that α = 1 / h (1 per hour) - a realistic value for many bacteria. In this case, the number of bacteria increases by a factor every hour e & approx; 2.71828 to. The following table shows the development over a few hours:

 time number 0 h 1 h 2 h 3 h 4 h 5 h 1000 2 718 7 389 20 086 54 598 148 413

You could easily check that with a calculator. Of course, the numbers have been rounded to whole values.

Let us consider the development over a somewhat larger period of time, namely over five days (where d stands for day - English day):

 time number 0 d 1 d 2 d 3 d 4 d 5 d 1000 26,5 · 1012 7,02 · 1023 1,86 · 1034 4,92 · 1044 1,3 · 1055

So, for example, after one day we already have 26.5 · 1012 Bacteria. Here I had to display the results with powers of ten, otherwise the display would be too long; for example is 1012 = 1 000 000 000 000, i.e. a 1 followed by 12 zeros (one trillion).

With exponential growth over a long enough period of time, the numbers are exorbitant, and emotional estimates can be extremely far off!

You can see that the number increases exorbitantly within a few days. Without the use of powers of ten, the length of the number representation would increase approximately linearly; with powers of ten you will see the same thing about the exponent. If you are not familiar with exponential growth, As a rule, long-term growth is underestimated massively. Surely after looking at the first table - with the development over a few hours - many readers would have thought that in the course of a few days it would be a few millions or billions, and not many trillions after just one day and far, far more after a few days. Anyone who has some experience with it will at least refrain from estimating emotionally and instead do a little math.

In practice, such dramatic growth is of course not possible in the long term, simply because so many bacteria would no longer fit in a test tube. For example, if each bacterium has a mass of one picogram (1 pg) - that is a trillionth of a gram - then a trillion bacteria (reached after approx. 34.5 days) together would already have a mass of one ton; this would require an unusually large test tube. Therefore, a realistic scenario is that growth is exponential at the beginning, but then decreases more or less drastically at some point. We'll look at the limits to growth below.

The (hypothetical) exponential development can also be shown with a diagram, here for the first 36 hours:

You can see that next to nothing seems to be doing for the first 30 hours: the blue curve stays very close to the y-axis (the bottom line). Then suddenly the post goes off: It looks as if growth is only suddenly starting here. But in reality it is relative Growth constant all the time: every bacterium behaved the same way from the beginning as it did later, and in every single hour their number increases by the same factor (here approx. 2.718). By the way, if the growth coefficient were only half as large (0.5 / h), we would simply have to wait twice as long to get the same growth.

If you draw the diagram with a logarithmic y-axis, you can follow the growth exactly from the beginning:

Note that the number of bacteria here always increases by a factor of 100 when you go from one dashed horizontal line to the next. In our example, this happens within approx. 4.6 hours.

In such a semi-logarithmic diagram (with logarithmic scaling only for the vertical, but not the horizontal axis) one always sees exponential growth as a straight line - this is a fundamental property of the exponential function. Because this linear progression is quite useful, this type of representation is widely used in science. You can also easily extrapolate the further growth graphically (advance into the future) under the assumption that it will continue to occur exponentially: You simply extend the straight line observed so far to the right. Of course, it is possible that the exponential growth will stop at some point, so the extrapolation will go wrong - more on the reasons for this later.

### Various questions

Certain questions often come up here:

### Is exponential growth at breakneck speed?

It is sometimes said that exponential growth is always incredibly fast. I wouldn't put it that way: At first it is quite slow. This is exactly why it is often very underestimated at the beginning. But the speed of growth then increases constantly and at some point becomes huge.

Of course, the size of the growth coefficient (the value of α in the above formulas) also plays a role. For example, this is relatively small when the world population grows, so that a significant increase can only be seen over entire generations - which of course still causes a number of serious problems.

### Why do the growth curves look so different?

As shown above, for a semi-logarithmic (semilogarithmic) diagram with exponential growth, a straight line is always obtained. With linear scaling of the axes, on the other hand, it is a curve that grows faster and faster, but one that is found with quite different shapes. For example, consider the following two graphs for bacterial growth:

Only in the second diagram do we have the impression that practically nothing happens for a long time and then suddenly things really get going. The difference is simply that the first diagram only shows the development over a short period of time, within which the growing size only increases by a factor of around 7 here. With the much longer period in the second diagram, the increase is much greater, which creates this special characteristic of the curve. At the beginning you don't see any increase at all because the relatively low values ​​appear negligibly small on a scale that still allows the presentation of the end result.

Quite a few deceptions - for example observed in the early days of the coronavirus crisis, frighteningly sometimes even among virologists - are essentially based on the fact that for some time only very low values ​​are perceived, the exponential growth of which is not taken into account (although one e.g. . could recognize it immediately with a meaningful graphical representation) and therefore feels safe. At some point there will be a nasty surprise.

### Why do you often come across logarithms here?

You have seen above that half-logarithmic diagrams are often used to represent exponential growth, and otherwise (e.g. in connection with the half-life, see below) a logarithm is encountered relatively often in this context. This is simply because the natural logarithm, for example, is the mathematical inverse of the natural exponential function. Similarly, the logarithm of ten is the inverse of the exponential function with base 10.

An example of this: If, with the exponential function described above, the time advances so far that αt increases by 17, the function increases by the factor exp (17) & approx; 24.1 million. Conversely, the natural logarithm of 24.1 million results in approx. 17. On the pocket calculator you will usually find a key labeled with “ln” - sometimes also “log”, but this is less clear; it could then, for example, also be the logarithm of ten, which is better called “lg” in order to avoid confusion.

### Can exponential growth go on indefinitely?

Mathematically, yes, but not in reality. Exponential growth will sooner or later reach its limits. What exactly happens then?

Let us consider the example of bacterial culture. They live on a nutrient solution and consume it more and more. Even if the nutrients were somehow constantly replaced and the waste products removed, sooner or later it would no longer be possible to keep the nutrient solution in such a state that the bacteria could continue to grow splendidly. So they inevitably come up against their limits because their livelihoods are being used up. It is also clear that this growth must stop well before the bacteria have the entire mass of the earth - and in our example this would be achieved fairly quickly with unlimited growth, namely within a good three and a half days.

It also happens in a similar way in many other contexts. When any size has grown sufficiently large through vigorous growth, it begins to have effects that often slow down further growth.

Mathematically, this is expressed by the fact that the growth coefficient no longer remains constant, but rather drops at some point.

### Exponential drop

Not only can growth occur exponentially, but also a decrease (a decrease). Mathematically, this is obtained when the growth coefficient becomes negative. In the case of exponential functions, the rule applies that the reciprocal of the original result is obtained by turning the exponent from positive to negative.

Example: e5 & approx; 148.4, e−5 & approx; 1 / 148.4 & approx; 0.00674.

As a physical example, consider the radioactive decay of cesium-137, one of the fission products of uranium that occurs in radioactive waste. So we start with a certain amount of it and see how this diminishes over time as these unstable atoms transform into other types of atoms while emitting radioactive radiation. For the individual atom this is a random (i.e. unpredictable) process, but for a large number of atoms one can calculate very well with average values. It is important to establish that the number of cesium atoms decaying per second is proportional to their number. Of course, the more you have of these atoms, the more of them decay per second. Because of that we have a negative rate of growth in their number that is proportional to that number itself. Mathematically, this leads to an exponential function again, only this time with a negative coefficient.

In the case of an exponential decrease, the so-called half-life is often givenT1/2 on - this is the time after which half of the original amount is still there. It can be easily calculated from the coefficient mentioned:

Here, ln 2? 0.693, the natural logarithm of 2.

In the case of exponential growth, you can do a Doubling time to calculate.

Of course, in reality (e.g. also in physics and technology) it happens very often that a negative growth rate is proportional to the current value of a decreasing quantity. Then there is always an exponential drop.

Of course, exponential decay also occurs at very different speeds. For example with radioactivity: While the half-life of some radioactive isotopes is a tiny fraction of a second, it is many billions of years for other isotopes.

### More examples of exponential growth or decline

In the following we consider a few more examples and their special circumstances as well as relatively easy to understand conclusions.

### Epidemics

An epidemic caused by infections with bacteria or viruses spreads through the transmission of these pathogens between people. Such a transmission process naturally becomes more likely in a population the more people are already infected. As long as the probability that an infected person will infect someone else remains constant and high enough, it will grow exponentially.

Often one looks at the so-called Reproduction factor - the average number of people infected by an infected person. Exponential growth occurs when the reproduction factor is greater than 1. On the other hand, there is an exponential decay if this factor is less than 1.

Contagion epidemics show a threshold behavior: If the probability of infection per infected person is pushed below a certain threshold, the cases of illness subside.

This leads to an interesting conclusion: in order to prevent a massive outbreak of an epidemic, one does not have to perfectly avoid contagion. It is enough to push the reproduction factor a little below 1 - which leads to a negative growth coefficient.

However, if the effort made is a little too weak, so that the reproductive factor is still above 1, the growth continues to be exponential - just a slower one. One cannot be satisfied with that if one wants to avoid the contagion of a large part of the population. This applies, for example, to the coronavirus pandemic that began in 2019: just increasing the number of intensive care places in hospitals could not solve the problem, as the load would sooner or later exceed any capacity limit with exponential growth. Such a strategy would be just as promising as fleeing from a shark by swimming quickly on the open sea: the shark is always faster than you.

Why exactly does an epidemic subside at some point even without active countermeasures?

Of course, even without active countermeasures, unlimited growth of an epidemic is not possible: More than 100% of a population can never be infected. Even before this is achieved, the reproductive factor and the rate of new infections decrease significantly, because the infected are surrounded by fewer and fewer uninfected people who could still be infected. A constant reproduction factor and a constant growth coefficient - the characteristics of exponential growth - are only possible as long as the majority of the population has not yet been infected. But in the end, most of them are caught - for no reason an epidemic does not end, as Donald Trump had to learn painfully in spring 2020 (especially for others).

The reproduction factor can also be reduced in other ways - some examples:

• One can reduce the likelihood of transmissions through reduced social contacts (“social distancing”).
• If there is a vaccination that can be used on a large scale, it may be possible to immunize so many people in time that the reproductive factor becomes less than 1 long before the disease has spread to a substantial part of the population.
• If the weather conditions change in such a way that this significantly reduces the number of infections, this sometimes means that a wave of disease subsides on its own.

In connection with the corona pandemic, it was often observed at the beginning that an understanding of the essential aspects of this crisis fails due to a lack of understanding of exponential growth. Even otherwise sensible people simply could not imagine that the increase in infections, which was still low at the time, would quickly lead to a major problem. That is probably one of the reasons why many people have misleading misinterpretations. There were, of course, a number of other reasons which I analyzed in my article on the detection of errors.

In detail, the conditions in epidemics are of course a lot more complicated than z. B. in the growth of bacteria in a nutrient solution. For example, the likelihood of infection within a family or a work group can be quite high, but lower across the boundaries of such groups. In addition, it is often difficult to calculate, since z. B. the number of those already infected is not so easy to determine. Dealing with these and many other problems as intelligently as possible is one of the central tasks of epidemiology.

### Viral dissemination of information

Similar to pathogens, information is transmitted from person to person.

Similar to pathogens, information can be transmitted from person to person. Such a thing can often also be described with statistical models. A reproductive factor can also be defined here - as the average number of people who are informed by one who has received the information. Once again, the information spreads exponentially when this reproduction factor is greater than 1, or exponentially when it is less than 1.

The reproduction factor depends on how exciting a message is, or more precisely how much it stimulates the recipient to spread it further. Of course, it also matters how easy it is for people to retransmit. Social media allow a particularly simple and fast distribution - often with just one or two mouse clicks.

“Viral spreading” of messages is extremely effective, but only works under special circumstances.

Only the smallest part of the messages reaches a reproduction factor above 1; in such cases one speaks of “viral” news or of viral distribution of the same. The number of people reached grows exponentially - slowly at first, then faster and faster. Such a wave of information usually only subsides after it has reached a large number of people and gradually runs out of the remaining number of people who have not yet been reached.

What is interesting is the insight that one is the widespread use of a Not viral message - for example with a reproduction factor of 0.8 - can hardly be enforced, for example by sending it to as many people as possible. Unless you are someone like Donald Trump, whom billions watch with admiration or horror, you will never reach nearly as many people as with viral distribution without the virality of your message.

Unfortunately, viral distribution is rarely successful with high-quality news, but rather for those who skillfully exploit any emotions with more or less dubious methods. For example, this occasionally succeeds with messages that give the impression that it is absolutely necessary to spread it in order to combat a sinister conspiracy. On the other hand, a text like this, which requires intensive reflection (far too exhausting for many people!), Has little chance of doing so. If you'd like to promote that anyway, you'll find suitable buttons for social media dissemination at the very bottom of the page!

### Interest on invested capital

When you lend someone money, you will usually receive interest in return. So you can gradually increase your capital in this way. If the interest rate is constant and you consume the interest received, the capital only grows linearly, i. H. every year by the same amount. However, if you always add the interest to the capital, you will in turn receive interest on this in the future (so-called compound interest), and the capital will grow exponentially. Unfortunately, it doesn't happen very quickly these days because the interest rates have gotten so low - unless you lend the money to someone who has to accept a high interest rate because your risk of default is so high.

Whether the interest rate is higher or lower than the inflation rate is pretty important in the long run!

If you want to take into account not only the nominal amount of money, but also the devaluation (inflation), you sometimes even get negative returns, at least with safe investments. In this sense, the wealth then gradually melts - with a constant rate of loss in an exponential manner.

More complicated with money than in many other growth phenomena is the fact that interest does not accrue continuously, but is paid at regular intervals. Therefore, the capital actually increases gradually and not really exponentially - but this does not play an essential role for the long-term view.

### Economic growth

Many economists assume that steady economic growth, ideally of several percent per year, is indispensable - not only for further increasing prosperity or overcoming poverty, but as a basic requirement for the stability of the entire economic system and our society.

Sustained economic growth sooner or later breaks all boundaries.

Constant economic growth would correspond to exponential growth in gross national product. This increase is usually accompanied by the increased consumption of resources, as well as the increased production of waste, pollutants and the like, and in particular also of the climate-damaging carbon dioxide. Obviously, sustained exponential growth would sooner or later lead to disaster, not to stability and lasting prosperity. We are already reaching the limits of growth in many places, since large parts of our economy are based on overexploitation and not on sustainability.

In principle, this dilemma could be resolved if prosperity on the one hand and resource consumption and environmental degradation on the other could be decoupled from one another - for example through improved energy efficiency. This decoupling has so far been somewhat successful, but gains in efficiency are mostly canceled out again in the end by additional consumption, which affects the impact on resources and the environment (→ rebound effect). So the challenge remains, either to decisively strengthen the decoupling mentioned and thus really achieve a limitation of resource consumption and environmental pollution, or to convert our economic system to a state without permanent growth. If this does not succeed, the growth will at some point be stopped unplanned by catastrophic processes.

### Atomic bomb and nuclear reactor

Atomic bombs provide the most impressive demonstration of the potential of exponential growth.

A nuclear chain reaction is used in an atom bomb (nuclear weapon), which leads to a massive exponential increase in the generated radiation and released energy over a period of around a few microseconds (1 μs). This is possible because the nuclear fission of certain types of atoms (isotopes) releases neutrons, which in turn can trigger the fission of other atoms they encounter. In this situation, the growth of neutron radiation and energy release is roughly exponential as long as enough fissile atoms remain available. The chain reaction ends well before all fissile atoms are split, as the explosion drives the material apart and thus reduces the likelihood that fissile atoms will still be hit by neutrons. However, splitting a few percent of the material is enough to cause an explosion with catastrophic consequences. Fortunately, it is technically not very easy to achieve the conversion of a few percent of the fissile material.

A hydrogen bomb is based on a chain reaction with nuclear fusion instead of nuclear fission, but this reaction is initiated by a nuclear fission bomb. In terms of quality, the above considerations also apply here, only that much more energy can be released.

How do you regulate the performance of a nuclear reactor - isn't that extremely critical?

A nuclear reactor for the production of nuclear energy is usually operated for a long time at a fairly precisely constant rate of the chain reaction in order to obtain a constant heat output. This means that the growth coefficient must be very close to zero so that there is neither growth nor decrease in performance. This is only possible through fast, active regulation that reacts very quickly to deviations from the desired performance by e.g. B. control rods more or less immersed in the reactor.

Incidentally, this regulation of the performance of a nuclear reactor would be extremely difficult in view of the rapid nuclear processes, if some of the neutrons from fission products (i.e. radioactive waste in the fuel rods) were not released with a certain delay, which makes the system much slower and thus more manageable. (The simple model with exponential growth therefore does not quite correspond to the somewhat more complex reality.) Other circumstances, such as the presence of certain transuranic elements or fission products, can also have a very detrimental effect on the controllability of a reactor. Therefore, depending on the type of reactor, certain operating states in which the control of criticality would no longer be guaranteed are strictly forbidden. The Chernobyl disaster was caused by the fact that such a ban was disregarded in the course of experiments (ironically with the aim of improving safety!). This resulted in an uncontrolled rapid increase in reactor output, which could no longer be stopped quickly enough to prevent the reactor and building from being destroyed by this criticality accident.

### laser

Exponential growth in light intensity in a laser can be used to generate extremely short and intense flashes of light.

A laser contains an arrangement of optical components (a laser resonator) between which light can travel back and forth continuously. In addition, this arrangement includes, for. B. a laser crystal, which can amplify light passing through if the energy supply is sufficiently strong (by “optical pumping”).

If this amplification is suddenly switched on while there is still almost no light in the resonator, this light is reduced in its power by z. B. increased a few percent. This leads to an exponential growth, which can lead to a very high performance in the course of a few hundred or a few thousand resonator revolutions. The increase in the power of the laser is stopped at some point, however, that the energy stored in the laser crystal (the basis for the amplification!) Is exhausted; the light output then drops rapidly again. This results in a possibly very intense laser pulse (flash of light), which in a very short time transports a large part of the energy that was previously stored in the laser crystal. The laser crystal basically serves as a type of energy storage device, which can be used for a comparatively long time (e.g.0.0001 seconds) and can release this energy again in a much shorter time.

The high speed of light allows extremely fast amplification even with a moderate amplification factor per revolution.

Due to the high speed of light, one revolution in the resonator only takes z. B. one nanosecond (one billionth of a second) or, in the case of a compact structure, even significantly less, and the duration of the light flash generated is correspondingly short: often only a few nanoseconds. During this time, however, the light is extremely intense - often enough to allow the material to evaporate immediately. Such laser pulses are widely used today, especially for laser material processing. However, so-called ones are also being used increasingly ultrashort Laser pulses with much shorter durations that are generated completely differently (i.e. not with the principle described above). Exponential amplification is often used for other purposes in order to really puff up these initially quite low-energy light pulses.

### Mechanical and other vibrations; Swinging of oscillators

Mechanical vibrations - such as a guitar string - often decay exponentially when no more energy is supplied. This has to do with the fact that the energy loss per period of oscillation is mostly proportional to the remaining energy of the oscillation to a good approximation. The same applies to many other types of damped oscillations, for example from electronic oscillating circuits.

Oscillators often start out with tiny oscillations that are then amplified exponentially.

Vibrations can also grow exponentially in some situations. Some oscillators contain a mechanism that supplies a vibratory system with energy during each period of oscillation, namely an amount that is initially (i.e. shortly after being switched on) proportional to the vibration energy that is already present. As you now know, that is exactly what leads to exponential growth. The mentioned proportionality naturally stops at some point when the oscillation becomes too strong; the available power as a prerequisite for the reinforcement is always limited. The vibrational energy then settles into a certain level.

If nothing is there yet, nothing can be amplified, can it?

Theoretically, one could think that such an oscillator will not start to oscillate by itself, because the oscillation energy is initially zero and then the mechanism mentioned cannot supply any energy. However, in reality, a tiny bit of energy is sufficient, for example. B. by the finest vibrations or thermal movement. Due to the exponential growth, it usually doesn't take too long for a strong oscillation to develop.

A laser (see above) is actually an example of such an oscillator - just not with a mechanical oscillation, but one of the electromagnetic field.

29.05.2020

In the media you hear repeatedly about the corona pandemic that the reproductive number has fallen to ONE, which supposedly would mean that an infected person would only infect a non-infected person. In my opinion this is not correct, because then you would have 2 infected people again on the next day and four on the next but one, etc. The reproduction number 1 means that the total number of infected people remains the same, i.e. for each newly infected person, one infected person would have to disappear, e.g. through recovery.

No, you misunderstood that. The size, the growth of which is considered here, is the number of new infections z. B. per day - not about the total number of those already infected. If the reproduction number is one, for example every infected person infects another person the following day (and then never again anyone), there would be the same number of new infections every day. The total number of infected people then naturally increases steadily - and in a linear fashion, if we continue to count those who have recovered and who have died among the infected.

It is similar in the example of nuclear fission in a nuclear reactor, which runs at constant power. Here there is a constant rate of nuclear fission (e.g. their number per second), which determines the thermal power output. The total number of split nuclei increases linearly, although the reproduction number is actually one.

29.03.2021

The graphics at the beginning of Corona were simply presented too roughly, leaving out the meaning of this presentation. With an R value of 2, would the infected always double every day? After 12 days after originally 1,000 people infected, around 4,000, etc., if you counted that way. These are all just theoretical values ​​that appear so rarely in reality, but are very well suited to spreading panic.

You have thoroughly misunderstood that: The R-value relates to a cycle of infection - the so-called generation time of four days - and not to a day.

In addition, it is not the case that you somehow calculate an R-value theoretically and then frighten people with a rapidly rising curve. On the contrary, one calls the actually increasing number of infections and then calculates the R-value from this. So it is not a prognosis, but first and foremost a quantity derived from observation data. This then results in a conditional prognosis, something like this: if If the R-value stays at this or that level, the infections will increase in a certain way. If this is unacceptable, it simply follows that the R-value has to be reduced - by taking suitable countermeasures.

Unfortunately, it happens again and again that a layman comes to nonsensical statements on the basis of a wrong understanding of the matter and then takes this as proof that all experts are obviously wrong.

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