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When a quantum system is coupled to its environment, a so-called open quantum system (OQS), it undergoes a very rapid decoherence process, one of the most fascinating and still poorly understood phenomena in fundamental physics.

Solving the dynamics of OQSs is challenging. It usually relies on a series of projection techniques and approximations, which aim to derive a quantum master equation for the reduced density matrix of the system. The resulting dynamics can be used to study dissipation and decoherence, the quantum-classical transition, and the emergence of irreversible behaviour, both at the quantum level and in the classical limit. 

In my group we study non-Markovian dynamics, occurring when there is no time-scale separation between environment and system, and the reduced dynamics are described in terms of memory kernels. Recent results show that in this case the decoherence process is substantially modified, with the emergence of what we call 'lateral coherences'  [Lally et al., Phys. Rev A (2022)].  We also study different types of environments, and their effects on the reduced dynamics obtained. We are now considering dissipative and fluctuating environments, and the implications of these on the decoherence process and the resulting thermodynamics. 

We also investigate the quantum-classical (QC) transition as a critical phenomenon. We adopt a Renormalization Group (RG) approach, which we extend to the case of OQS, to identify the scaling properties of the reduced dynamics. Recent findings suggest that together with quantum dissipation, the low-frequency behaviour of the decoherence process too needs to be taken into account to characterize fully the QC transition. The obtained RG flow, driven by a generalized Wegner-Houghton equation, describes the progressive elimination of quantum fluctuations, and therefore should admit the classical case as a fixed point.  

OQS can be used as well to study the breakdown of time-reversal symmetry in reduced dynamics. In my group, we have shown that a careful derivation of reduced OQS dynamics reproduces irreversible behaviour in terms of monotonically time-increasing classical and quantum entropies in the Markov limit, but – in contrast to expectations – the reduced macroscopic equations of motion for the system of interest still show time-reversal invariance. This happens with respect to time t = 0, at which the system is Markovianized: As a result, the system can thermalize in two opposing directions of time, emerging symmetrically from t = 0, and it shows irreversible behaviour both towards the future and towards the past [Guff et al., Scientific Reports, 2025]. 

Molecular bio-systems, such as gene regulatory networks, are affected by stochastic fluctuations.

Fluctuations due to low copy numbers of proteins, or to gene expression bursts, define so-called 'intrinsic' noise, while fluctuations originating in the environment are usually referred to as 'extrinsic' noise. This part of my research concerns the study of both intrinsic and extrinsic noise, and of their interplay, by using Langevin and Fokker-Planck approaches. In particular, extrinsic noise can have highly non-trivial and counterintuitive effects, which may lead to noise-induced transitions [Rocco et al., Springer (2013)], well known in a variety of physical and chemical systems. These effects are related to the so-called Stratonovich drift, which arises when noise is gaussian and white, and may lead to a change of the stability and numbers of steady states of the corresponding deterministic system.  Consideration of these effects has led to the introduction of the concept of stochastic control in metabolic networks [Rocco, Phys. Biol. (2009)], where noise itself is proven to be able to act as a control mechanism that tunes metabolic concentrations and fluxes.

In my group we are now developing a novel dynamical framework which describes bounded, non-gaussian, and nonlinear noise in the case of slow extrinsic fluctuations in gene expression [Aquino & Rocco, Math. Biosci. Eng. (2020)]. These fluctuations also lead to noise induced transitions, albeit in a  different manner with respect to those emerging when the noise is gaussian and white. For instance, we have shown that they can account for the growth and drug tolerance heterogeneity observed in clonal populations of bacterial cells [Rocco et al., PLOS One (2103); Hingley-Wilson et al., PNAS (2020)]. 

Recently we have succeeded in developing a formalism which allows us to describe extrinsic fluctuations in gene expression which are slow, but not infinitely slower than transcriptional and translational dynamics [Aquino & Rocco, Phys. Rev. E (2023)].

A major problem in developmental biology is to explain the emergence of different cell types from multipotent stem cells.

The problem is exquisitely suitable for the mathematical framework of Dynamical Systems Theory, which naturally leads to description and understanding of cellular decision-making processes in terms of bifurcation analysis. In collaboration with experimentalists at the University of Bath, we have constructed the first core gene regulatory network describing stable differentiation of melanocytes in zebrafish [Greenhill et al., PLoS Genetics (2011)]. By using a combination of mathematical analysis and simulations, we could predict mathematically and validate experimentally several new features of this network. More recently we have refined that core network, to include an analysis of the role played by Wnt signaling in melanocyte differentiation [Vibert et al., Pigment Cell & Melanoma Research (2017)], and extended it to other neural crest derived pigment cells [Petratou et al., PLoS Genetics (2018)].

Currently we are investigating the relevance of cyclic dynamics in the differentiation process. We have introduced a new framework [Kelsh et al., Development (2021)] in which the multipotent cell is cycling through a series of metastable states, and while sojourning in any of them, may be exposed to external signals, which break the cycle and drive further differentiation [Farjami et al., J. Royal Soc. Interface (2021)]. Experimental evidence, obtained by analysing in detail direct vs progressive fate restriction models of pigment cells in zebrafish is consistent with this hypothesis [Subkhankulova et al., Nat. Comm. (2023)]. 

We are now investigating the relevance of stochastic fluctuations in this novel framework.