A deterministic system is non-stochastic. It is not subject to change. A stochastic system is probabilistic. It can change with calculable probability.

A system is a system. This is neither deterministic nor stochastic. However, if we want describe the development of a (dynamic) system, we use a model, and such a model (description) can be deterministic or stochastic.

A deterministic model can eventually be given as a mathematic formula or equation (or a set of equations, e.g. differential equations). It allows us to assume we know everything (relevant) that happens in the system and that this is correctly specified in the formula(s). For any set of parameters the entire history (past and future) of the system is thus "known", as we can directly evaluate the formula(s) for any given time-point (that can be practically quite demanding, even impossible, but we look at the principles here).

A stochastic model is used if we can not (or don't want to) model quantitative relationships between the components of the system but instead can (or want to) give only probabilities for some events happening during some (usually short) periods of time. Having some starting values we can find probabilities of the system being in diffenet possible future states.

Example: diffusion. Say you put a drop of microbes onto a wet surface. A deterministic model would be a formula giving the concentration of microbes at any distance of the drop centre at any time. A stochastic model would be based on the movement of the individual microbes, what is modelled as a random walk: during a small time-step, each microbe moves a tiny step in a random direction according to a probability distribution. To see how the concentration in a distance is after some time, one needs to run the model, using random values generated by a random number generator (RNG) according to the desired distribution. This will result in one possible outcome, from which the concentration can be determined (e.g. as number of microbes in a small area in the given distance from the origin). Reapeating this many times (and assuming that the frequency distribution of the values from the RNG will approximate the shape of the desired probability distribution), we get a frequency distribution of concentrations that we can interpret as a probability distribution, and we can eventually say that, based on our model, we expect the concentration to be in some range with some given probability.

In addition to Jochem's answer. In simplified manner, Y = a + bX is deterministic or mathematical while Y = a + bX + U is stochastic. The difference is the error or stochastic term in the model

Sorry Job, but I think you confuse something here.

What you present is best described as a statistical model, which has a deterministic part (a+bX) and a stochastic part (U), if you wish. But that is a dissection that works only in special cases. More generally, a statistical model relates -deterministacally- values of one or serveral predictors to a response variable. The response variable is described as a random variable with a probability distribution conditional on the predictor values.

A stochastic model would rather model that we are not so sure how large a or b is in a particular realisation. In this case, this can always be expressed eventually as a statistical model, getting an analytical solution for the conditional distribution of a+bX. If we would model this stochastically, we would evaluate the model for an ensemble or different values for a and b (all drawn from a distribution according to our assumptions) and record an ensemble of different values for a+bX, and we would use the observed frequency distribution as en estimate for the conditional probability distribution of the response.

Just found two nice links that may help clarifying the issue:

http://www.dcs.gla.ac.uk/~srogers/teaching/mscbioinfo/SysBio2.pdf

http://www.inrialpes.fr/schoolleshouches07/pres/Gonze_LesHouches_1.pdf

You have defined it perfectly already. A deterministic system is a system in which no randomness is involved in the development of future states of the system. A stochastic system has a random probability distribution or pattern that may be analysed statistically but may not be predicted precisely :)

I wouldn't oppose Job's example as a stochastic one (despite the fact that the quantity U is not stated as a random variable - probably by simple forgetting to be more precise when writing to public readers:). Namely, even Jochen's requirement " A stochastic model would rather model that we are not so sure how large a or b is in a particular realisation. " is fulfilled. Namely, a+U represents our lack of knowledge how large a is. Moreover, Job's example can be seen as a dynamical system if X can be interpreted as the time variable: It describes a motion with constant speed with random initial position.

Regards, Joachim

PS. I think that strict definition distinguishing between deterministic and stochastic systems cannot be given, since e.g. Markov processes can be seen as a system describing deterministic evolution of the probabilities, which, however, describe stochastically the position in the state space.

The usual assumption about U is assumed here. This is a mature community. The assumption of NID (0, e square) holds.

@ Job

I am trying to decipher the abbreviation as "Normal Independent ...?.. " which impliess a furthe guess that you are suggesting perturbation by normal white noise, which makes sense for discrete time only. Then, indeed, Jochens suggestion that this is statistical model becomes a better justification. For continuous time, the reason is that the perturbation by a normal (gaussian) process with covariance function R(t,s) = 0 for s\ne t t,s \in R, is not accepted as a right model for applications (basically, due to very irregular properties). Note, that this is not equivalent to the closest model of the solutions to the Ito SDE dX(t) = b dt + dW(t), where W stands for some Brownian motion. In this case X(t) = X(0) + b t + W(t), where the covariance of W(t) is min{s,t} (not equal 0 for different s and t.

Regards, Joachim

https://books.google.dz/books?id=CQVZAgAAQBAJ&pg=PR10&lpg=PR10&dq=Difference+between+Stochastic+and+Deterministic+Systems+(Mathematically-Physically)?&source=bl&ots=zOBZW7yJqt&sig=0p6Jf8SL8ztQbr-nOLpSpUuDGxA&hl=fr&sa=X&ved=0ahUKEwj-w9yj6_3ZAhVMjiwKHRB2A48Q6AEIWjAL#v=onepage&q=Difference%20between%20Stochastic%20and%20Deterministic%20Systems%20(Mathematically-Physically)%3F&f=false

Hi

this is a good question!

I derived a new formulation for analysis of natural phenomena, called the state based philosophy (SBP). In the SBP there is no need to divide into the deterministic and the stochastic, both are treated with the same formulation, which needs few reliable data points for calibration. To this end the SBP is validated in decision making (quantile methods), fragility analysis, engineering design, probability and etc.

look for the SBP in the researchgate and in the open literature.

A stochastic system is a system whose future states, due to its components' possible interactions, are not known precisely. A system modeler does not precisely know the possible coalition and how the behaviors will emerge. This is the case in non-deterministic systems formed through the collective dynamics of participating components. However, a composed system is fundamentally stochastic/non-deterministic as it does not have full control over all the operations of cooperating elements.

I suggest to read our researches :

Article Chaos in a stochastic cancer model

Conference Paper A Stochastic Model for Chaotic-Weather Prediction based on H...

Hi

Difference between deterministic and stochastic models are mathematical but not physical! Deterministic models based on solution of equations that are in turn based on assumptions (unreliable). In most cases deterministic models contain epistemic uncertainty. On the other hands stochastic models based on experiments (reliable) at first level. On the second level, assumed probability functions are fitted on the data (little uncertainty). Overall stochastic models are more reliable than the deterministic models. Recently our research team based on logical reasoning derived a method that is most reliable and is a replacement for both! The proposed method is called the change of state philosophy which is digested in the Persian Curve. Look for the proposed method in the literature?

In terms of population density, For small population stochastic model gives better results as compare to deterministic model. but for large population stochastic model converges to deterministic model.