How bad is it if we don't know the distribution of an average?

Let's assume that it takes on average $$\langle W\rangle$$ work to perform some process.

While we do know that the fluctuations i.e. the difference between single realizations of the process $$W_1$$,$$W_2$$,... and the average $$\langle W\rangle$$ are significant we have no idea how big this difference actually is.

They could be far or very far from the average, all we know is that they cannot be ignored.

In other words we don't know how the probability distribution looks like with which one could calculate $$\langle W\rangle$$ .

Now my question: How bad is that? Does $$\langle W\rangle$$ have any significant meaning in this context? Or is it then completeley useless?

Bonus info:

More exactly I am talking about the Two-projective-measurment (TPM) scheme. The goal of said protocol is to define the work (in a thermodynamically isolated system) of a process done in the quantum regime as the difference of two measurement outcomes:

$$W_{ij} = E_i - E_j\tag{1}$$

With this one obtains a probability distribution of work $$P(W)$$ with which he/she then can calculate the average $$\langle W\rangle_{TPM}$$.

Simultaneously we can (still in a thermodynamically isolated system) define the work in a unitary process as:

$$\langle W\rangle_{avg} = \mathrm{Tr}(\rho_i H_0) - \mathrm{Tr}(\rho_f H_0)\tag{2}$$

where $$\rho_f = U \rho_i U^{\dagger}\tag{3}$$

Now the whole point is that $$\langle W\rangle_{TPM} = \langle W\rangle_{avg}\tag{4}$$

only if the initial state is a classical state i.e. that has no coherences.

That way we are able to obtain a work distribution from the TPM scheme for a unitary process (for thermally isolated system) if the system is initially classical.

But in fact nothing is stopping us from calculating $$\langle W\rangle_{avg}$$ from eq (2) for a system which is initially not classical. Thus we can calculate an average value $$\langle W\rangle_{avg}$$ for system with coherences it is just that we have no distribution from the TPM scheme as

$$\langle W\rangle_{avg} \neq \langle W\rangle_{TPM}$$

for initially non-classical systems.

Thus, we have an average $$\langle W\rangle_{avg}$$ from eq (2) but no information about the distribution.

And in fact it has been shown that it is impossible to obtain said distribution for coherent processes.

So I ask again: How bad is it to have an average value $$\langle W\rangle_{avg}$$ from eq. (2) but no corresponding distribution which would allow us to quantify the fluctuations that occur there (and we know fluctuations are relevant in the Quantum regime).

• I think this question needs some more context. Are you sure you have no constraints on $\sigma_W$ at all? If you have no idea whether the work is $10^{-10}W$, or $W$, or $10^{10}W$, it might be pretty bad. – pela Feb 5 at 16:57
• maybe you are asking about maximum entropy methods, see en.wikipedia.org/wiki/Principle_of_maximum_entropy? – hyportnex Feb 5 at 17:19
• @pela added info – CatoMaths Feb 5 at 18:28