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We relate the probability distribution of the work done on a statisticalsystem under a sudden quench to the Lanczos coefficients corresponding toevolution under the post-quench Hamiltonian. Using the general relation betweenthe moments and the cumulants of the probability distribution, we show that theLanczos coefficients can be identified with physical quantities associated withthe distribution, e.g., the average work done on the system, its variance, aswell as the higher order cumulants. In a sense this gives an interpretation ofthe Lanczos coefficients in terms of experimentally measurable quantities.Consequently, our approach provides a way towards understanding spreadcomplexity, a quantity that measures the spread of an initial state with timein the Krylov basis generated by the post quench Hamiltonian, from athermodynamical perspective. We illustrate these relations with two examples.The first one involves quench done on a harmonic chain with periodic boundaryconditions and with nearest neighbour interactions. As a second example, weconsider mass quench in a free bosonic field theory in $d$ spatial dimensionsin the limit of large system size. In both cases, we find out the timeevolution of the spread complexity after the quench, and relate the Lanczoscoefficients with the cumulants of the distribution of the work done on thesystem.

Time evolution of spread complexity and statistics of work done in quantum quenches