Roderick C. Dewar

Maximum entropy production and the fluctuation theorem

Recently the author used an information theoretical formulation of non-equilibrium statistical mechanics (MaxEnt) to derive the fluctuation theorem (FT) concerning the probability of second law violating phase-space paths. A less rigorous argument leading to the variational principle of maximum entropy production (MEP) was also given. Here a more rigorous and general mathematical derivation of MEP from MaxEnt is presented, and the relationship between MEP and the FT is thereby clarified. Specifically, it is shown that the FT allows a general orthogonality property of maximum information entropy to be extended to entropy production itself, from which MEP then follows. The new derivation highlights MEP and the FT as generic properties of MaxEnt probability distributions involving anti-symmetric constraints, independently of any physical interpretation. Physically, MEP applies to the entropy production of those macroscopic fluxes that are free to vary under the imposed constraints, and corresponds to selection of the most probable macroscopic flux configuration. In special cases MaxEnt also leads to various upper bound transport principles. The relationship between MaxEnt and previous theories of irreversible processes due to Onsager, Prigogine and Ziegler is also clarified in the light of these results.

Modeling carbon allocation in trees: a search for principles

We review approaches to predicting carbon and nitrogen allocation in forest models in terms of their underlying assumptions and their resulting strengths and limitations. Empirical and allometric methods are easily developed and computationally efficient, but lack the power of evolution-based approaches to explain and predict multifaceted effects of environmental variability and climate change. In evolution-based methods, allocation is usually determined by maximization of a fitness proxy, either in a fixed environment, which we call optimal response (OR) models, or including the feedback of an individuals strategy on its environment (game-theoretical optimization, GTO). Optimal response models can predict allocation in single trees and stands when there is significant competition only for one resource. Game-theoretical optimization can be used to account for additional dimensions of competition, e.g., when strong root competition boosts root allocation at the expense of wood production. However, we demonstrate that an OR model predicts similar allocation to a GTO model under the root-competitive conditions reported in free-air carbon dioxide enrichment (FACE) experiments. The most evolutionarily realistic approach is adaptive dynamics (AD) where the allocation strategy arises from eco-evolutionary dynamics of populations instead of a fitness proxy. We also discuss emerging entropy-based approaches that offer an alternative thermodynamic perspective on allocation, in which fitness proxies are replaced by entropy or entropy production. To help develop allocation models further, the value of wide-ranging datasets, such as FLUXNET, could be greatly enhanced by ancillary measurements of driving variables, such as water and soil nitrogen availability.

Why is plant-growth response to elevated CO2 amplified when water is limiting, but reduced when nitrogen is limiting? A growth-optimisation hypothesis

Experimental evidence indicates that the stomatal conductance and nitrogen concentration ([N]) of foliage decline under CO2 enrichment, and that the percentage growth response to elevated CO2 is amplified under water limitation, but reduced under nitrogen limitation. We advance simple explanations for these responses based on an optimisation hypothesis applied to a simple model of the annual carbon-nitrogen-water economy of trees growing at a CO2-enrichment experiment at Oak Ridge, Tennessee, USA. The model is shown to have an optimum for leaf [N], stomatal conductance and leaf area index (LAI), where annual plant productivity is maximised. The optimisation is represented in terms of a trade-off between LAI and stomatal conductance, constrained by water supply, and between LAI and leaf [N], constrained by N supply. At elevated CO2 the optimum shifts to reduced stomatal conductance and leaf [N] and enhanced LAI. The model is applied to years with contrasting rainfall and N uptake. The predicted growth response to elevated CO2 is greatest in a dry, high-N year and is reduced in a wet, low-N year. The underlying physiological explanation for this contrast in the effects of water versus nitrogen limitation is that leaf photosynthesis is more sensitive to CO2 concentration ([CO2]) at lower stomatal conductance and is less sensitive to [CO2] at lower leaf [N].

Statistical mechanics unifies different ecological patterns.

Recently there has been growing interest in the use of maximum relative entropy (MaxREnt) as a tool for statistical inference in ecology. In contrast, here we propose MaxREnt as a tool for applying statistical mechanics to ecology. We use MaxREnt to explain and predict species abundance patterns in ecological communities in terms of the most probable behaviour under given environmental constraints, in the same way that statistical mechanics explains and predicts the behaviour of thermodynamic systems. We show that MaxREnt unifies a number of different ecological patterns: (i) at relatively local scales a unimodal biodiversity-productivity relationship is predicted in good agreement with published data on grassland communities, (ii) the predicted relative frequency of rare vs. abundant species is very similar to the empirical lognormal distribution, (iii) both neutral and non-neutral species abundance patterns are explained, (iv) on larger scales a monotonic biodiversity-productivity relationship is predicted in agreement with the species-energy law, (v) energetic equivalence and power law self-thinning behaviour are predicted in resource-rich communities. We identify mathematical similarities between these ecological patterns and the behaviour of thermodynamic systems, and conclude that the explanation of ecological patterns is not unique to ecology but rather reflects the generic statistical behaviour of complex systems with many degrees of freedom under very general types of environmental constraints.

Optimal function explains forest responses to global change

Plant responses to global changes in carbon dioxide (CO2), nitrogen, and water availability are critical to future atmospheric CO2 concentrations, hydrology, and hence climate. Our understanding of those responses is incomplete, however. Multiple-resource manipulation experiments and empirical observations have revealed a diversity of responses, as well as some consistent patterns. But vegetation models—currently dominated by complex numerical simulation models—have yet to achieve a consensus among their predicted responses, let alone offer a coherent explanation of the observed ones. Here we propose an alternative approach based on relatively simple optimization models (OMs). We highlight the results of three recent forest OMs, which together explain a remarkable range of observed forest responses to altered resource availability. We conclude that OMs now offer a simple yet powerful approach to predicting the responses of forests—and, potentially, other plant types—to global change. We recommend ways in which OMs could be developed further in this direction.

A single‐substrate model to interpret intra‐annual stable isotope signals in tree‐ring cellulose

The carbon and oxygen stable isotope composition of wood cellulose (delta(13)C(cellulose) and delta(18)O(cellulose), respectively) reveal well-defined seasonal variations that contain valuable records of past climate, leaf gas exchange and carbon allocation dynamics within the trees. Here, we present a single-substrate model for wood growth to interpret seasonal isotopic signals collected in an even-aged maritime pine plantation growing in South-west France, where climate, soil and flux variables were also monitored. Observed seasonal patterns in delta(13)C(cellulose) and delta(18)O(cellulose) were different between years and individuals, and mostly captured by the model, suggesting that the single-substrate hypothesis is a good approximation for tree ring studies on Pinus pinaster, at least for the environmental conditions covered by this study. A sensitivity analysis revealed that the model was mostly affected by five isotopic discrimination factors and two leaf gas-exchange parameters. Modelled early wood signals were also very sensitive to the date when cell wall thickening begins (t(wt)). Our model could therefore be used to reconstruct t(wt) time series and improve our understanding of how climate influences this key parameter of xylogenesis.

Maximum Entropy Production as an Inference Algorithm that Translates Physical Assumptions into Macroscopic Predictions: Don’t Shoot the Messenger

Is Maximum Entropy Production (MEP) a physical principle? In this paper I tentatively suggest it is not, on the basis that MEP is equivalent to Jaynes’ Maximum Entropy (MaxEnt) inference algorithm that passively translates physical assumptions into macroscopic predictions, as applied to non-equilibrium systems. MaxEnt itself has no physical content; disagreement between MaxEnt predictions and experiment falsifies the physical assumptions, not MaxEnt. While it remains to be shown rigorously that MEP is indeed equivalent to MaxEnt for systems arbitrarily far from equilibrium, work in progress tentatively supports this conclusion. In terms of its role within non-equilibrium statistical mechanics, MEP might then be better understood as Messenger of Essential Physics.

Maximum entropy production and plant optimization theories

Plant ecologists have proposed a variety of optimization theories to explain the adaptive behaviour and evolution of plants from the perspective of natural selection (‘survival of the fittest’). Optimization theories identify some objective function—such as shoot or canopy photosynthesis, or growth rate—which is maximized with respect to one or more plant functional traits. However, the link between these objective functions and individual plant fitness is seldom quantified and there remains some uncertainty about the most appropriate choice of objective function to use. Here, plants are viewed from an alternative thermodynamic perspective, as members of a wider class of non-equilibrium systems for which maximum entropy production (MEP) has been proposed as a common theoretical principle. I show how MEP unifies different plant optimization theories that have been proposed previously on the basis of ad hoc measures of individual fitness—the different objective functions of these theories emerge as examples of entropy production on different spatio-temporal scales. The proposed statistical explanation of MEP, that states of MEP are by far the most probable ones, suggests a new and extended paradigm for biological evolution—‘survival of the likeliest’—which applies from biomacromolecules to ecosystems, not just to individuals.

Drought‐related tree mortality: addressing the gaps in understanding and prediction

Increased tree mortality during and after drought has become a research focus in recent years. This focus has been driven by: the realisation that drought-related tree mortality is more widespread than previously thought; the predicted increase in the frequency of climate extremes this century; and the recognition that current vegetation models do not predict drought-related tree mortality and forest dieback well despite the large potential effects of these processes on species composition and biogeochemical cycling. To date, the emphasis has been on understanding the causal mechanisms of drought-related tree mortality, and on mechanistic models of plant function and vegetation dynamics, but a consensus on those mechanisms has yet to emerge. In order to generate new hypotheses and to help advance the modelling of vegetation dynamics in the face of incomplete mechanistic understanding, we suggest that general patterns should be distilled from the diverse and as-yet inconclusive results of existing studies, and more use should be made of optimisation and probabilistic modelling approaches that have been successfully applied elsewhere in plant ecology. The outcome should inform new empirical studies of tree mortality, help improve its prediction and reduce model complexity.

Why does leaf nitrogen decline within tree canopies less rapidly than light? An explanation from optimization subject to a lower bound on leaf mass per area

A long-established theoretical result states that, for a given total canopy nitrogen (N) content, canopy photosynthesis is maximized when the within-canopy gradient in leaf N per unit area (N(a)) is equal to the light gradient. However, it is widely observed that N(a) declines less rapidly than light in real plant canopies. Here we show that this general observation can be explained by optimal leaf acclimation to light subject to a lower-bound constraint on the leaf mass per area (m(a)). Using a simple model of the carbon-nitrogen (C-N) balance of trees with a steady-state canopy, we implement this constraint within the framework of the MAXX optimization hypothesis that maximizes net canopy C export. Virtually all canopy traits predicted by MAXX (leaf N gradient, leaf N concentration, leaf photosynthetic capacity, canopy N content, leaf-area index) are in close agreement with the values observed in a mature stand of Norway spruce trees (Picea abies L. Karst.). An alternative upper-bound constraint on leaf photosynthetic capacity (A(sat)) does not reproduce the canopy traits of this stand. MAXX subject to a lower bound on m(a) is also qualitatively consistent with co-variations in leaf N gradient, m(a) and A(sat) observed across a range of temperate and tropical tree species. Our study highlights the key role of constraints in optimization models of plant function.