We study the coarse-graining approach to derive a generator for the evolution of an open quantum system over a finite time interval. The approach does not require a secular approximation but nevertheless generally leads to a Lindblad–Gorini–Kossakowski–Sudarshan generator. By combining the formalism with full counting statistics, we can demonstrate a consistent thermodynamic framework, once the switching work required for the coupling and decoupling with the reservoir is included. Particularly, we can write the second law in standard form, with the only difference that heat currents must be defined with respect to the reservoir. We exemplify our findings with simple but pedagogical examples.

A set of core features is set forth as the essence of a thermodynamic description, which derive from large-deviation properties in systems with hierarchies of timescales, but which are not dependent upon conservation laws or microscopic reversibility in the substrate hosting the process. The most fundamental elements are the concept of a macrostate in relation to the large-deviation entropy, and the decomposition of contributions to irreversibility among interacting subsystems, which is the origin of the dependence on a concept of heat in both classical and stochastic thermodynamics. A natural decomposition that is known to exist, into a relative entropy and a housekeeping entropy rate, is taken here to define respectively the <i>intensive</i> thermodynamics of a system and an <i>extensive</i> thermodynamic vector embedding the system in its context. Both intensive and extensive components are functions of Hartley information of the momentary system stationary state, which is information about the joint effect of system processes on its contribution to irreversibility. Results are derived for stochastic chemical reaction networks, including a Legendre duality for the housekeeping entropy rate to thermodynamically characterize fully-irreversible processes on an equal footing with those at the opposite limit of detailed-balance. The work is meant to encourage development of inherent thermodynamic descriptions for rule-based systems and the living state, which are not conceived as reductive explanations to heat flows.

While the fundamental laws of physics are time-reversal invariant, most macroscopic processes are irreversible. Given that the fundamental laws are taken to underpin all other processes, how can the fundamental time-symmetry be reconciled with the asymmetry manifest elsewhere? In statistical mechanics (SM), progress can be made with this question. What I dub the 'Zwanzig-Zeh-Wallace framework' can be used to construct the irreversible equations of SM from the underlying microdynamics. Yet this framework uses coarse-graining, a procedure that has faced much criticism. I focus on two objections in the literature: claims that coarse-graining makes time-asymmetry (i) 'illusory' and (ii) 'anthropocentric'. I argue that these objections arise from an unsatisfactory justification of coarse-graining prevalent in the literature, rather than from coarse-graining itself. This justification relies on the idea of measurement imprecision. By considering the role that abstraction and autonomy play, I provide an alternative justification and offer replies to the illusory and anthropocentric objections. Finally, I consider the broader consequences of this alternative justification: the connection to debates about inter-theoretic reduction and the implication that the time-asymmetry in SM is weakly emergent. 1Introduction 1.1Prospectus2The Zwanzig-Zeh-Wallace Framework3Why Does This Method Work? 3.1The special conditions account3.2When is a density forwards-compatible?4Anthropocentrism and Illusion: Two Objections 4.1The two objections in more detail4.2Against the justification by measurement imprecision5An Alternative Justification 5.1Abstraction and autonomy5.2An illustration: the Game of Life6Reply to Illusory7Reply to Anthropocentric8The Wider Landscape: Concluding Remarks 8.1Inter-theoretic relations8.2The nature of irreversibility. However, there is also an ensemble variant of this description. Here probability densities over Γ -space, ρ , evolve according to Liouville’s equation, which, like Hamilton’s equations, is TRI. 2 Stage 2: The concept of coarse-graining was originally introduced in a specific form by Gibbs ([ 1903 ]) which I first recall, before describing the generalized coarse-graining projections used by the ZZW framework. Gibbs proposes that the accessible phase-space Γ is partitioned into small, finite volume elements Δ V m . The coarse-grained density ρ c g ( q , p ) is then defined by averaging the original probability density ρ ( q , p ) in each of these boxes. So coarse-graining throws away the information about how exactly the ensemble is distributed across each box. Gibbs describes the evolution of the probability density by analogy with an ink drop. Dropping blue ink into a glass of water results in the whole glass appearing light blue. However, a drop of ink is an incompressible fluid and so its volume is constant. Upon examination under a microscope, we would see Below are three examples of a coarse-graining projection P ^ defining a relevant density ρ r . In these examples, the density is defined over a reduced number of degrees of freedom of the systems. Hence we speak of ‘relevant degrees of freedom’, as well as ‘relevant densities’. The archetypal Gibbsian coarse-graining discussed above can be written as a projection, P ^ c g . P ^ c g averages over small, finite volume elements Δ V m ( m = 1 , 2 … ) that cover the 6 N -dimensional phase space Γ . These volume elements Δ V m are sometimes referred to as ‘coarse-grained boxes’ or ‘cells of a partition’. In addition, the partition is chosen by us: ‘the occurrence and direction of a temporal change of the entropy [ … ] depends essentially on our human choice of the size of the finite equal cells of boxes into which we partition [ … ] phase space’ ( Grünbaum [1973] , p. 647). The objection extends to all instances of P ^ ; ‘a Zwanzig projection (describing generalized coarse-graining) can be arbitrarily chosen for convenience’ ( Zeh [2007] , p. 67) Grünbaum ([ 1973 ]) points out that the charge of anthropocentrism here differs from the more general claim that scientific theories are human constructs. Thus, we can never locate a system precisely in phase space; we only know p and q to a certain degree of accuracy. The cells over which we average with the P ^ c g for the archetypal Gibbsian coarse-graining have a size that corresponds to ‘the limits of accuracy actually available to us’ ( Tolman [1938] , p. 167). Because we could never, ex hypothesi , measure the system accurately enough, we are unable to distinguish between the coarse and fine-grained distributions ρ and ρ r . Thus, according to this MI justification, the answer to choice is that we must pick the coarse-graining P ^ that matches our observational capacities. A similar argument arises in the Boltzmannian approach to SM, where phase space is partitioned into ‘macrostates’. Every microstate corresponds to one macrostate. A particular macrostate is defined by values of macrovariables, such as volume, temperature and pressure. These macrostates are sets of microstates that are ‘empirically indistinguishable’. Thus, an appeal is once again made to our observational capacities. 14 The illusory and anthropocentric objections arise from this justification of coarse-graining (rather than coarse-graining itself). Hence, coarse-graining does not lead to a specific anthropocentrism (which one might have been concerned would render SM incompatible with scientific realism). However, as discussed in Section 5, different levels of description are useful for different purposes and what is deemed useful may be relative to our human interests. Here our measuring capacities and imprecision are certainly relevant. Were we the size of a Maxwell demon and endowed with an ability to manipulate gas molecules, violations of the second law of thermodynamics might be expected. From their microscopic perspective, the second law might not seem like an obvious regularity in nature.

result represents coarse graining—i.e., information loss by smoothing out very fine-scale detail. Some caveats should be considered with the above. 1. Like In physics, maximum entropy thermodynamics (colloquially, MaxEnt thermodynamics) views equilibrium thermodynamics and statistical mechanics as inference processes. More specifically, MaxEnt applies inference techniques rooted in Shannon information theory, Bayesian probability, and the principle of maximum entropy. These techniques are relevant to any situation requiring prediction from incomplete However, as time evolves, that initial information we had becomes less directly accessible. Instead of being easily summarizable in the macroscopic description of the system, it increasingly relates to very subtle correlations between the positions and momenta of individual molecules. (Compare to Boltzmann's H-theorem.) Equivalently, it means that the probability distribution for the whole system, in 6N-dimensional phase space, becomes increasingly irregular, spreading out into long thin fingers rather than the initial tightly defined volume of possibilities. Classical thermodynamics is built on the assumption that entropy is a state function of the macroscopic variables—i.e., that none of the history of the system matters, so that it can all be ignored. The extended, wispy, evolved probability distribution, which still has the initial Shannon entropy STh(1), should reproduce the expectation values of the observed macroscopic variables at time t2. However it will no longer necessarily be a maximum entropy distribution for that new macroscopic description. On the other hand, the new thermodynamic entropy STh(2) assuredly will measure the maximum entropy distribution, by construction. Therefore, we expect: