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Predstavljamo projekt izgradnje istraživačkih kapaciteta u MEMS tehnologiji u Hrvatskoj kroz nedavno odobreni projekt financiran iz Europskih strukturnih fondova, zasnovan na suradnji između Sveučilišta i tvrtke u kategoriji malog i srednjeg poduzetništva.

Are living entities optimized to produce as little or as much entropy as possible? We show here that at least one important molecular motor – the ATP synthase – evolved in accord with the statistical selection principle of Maximum Shannon Entropy (MaxEnt) and one of its corrolaries – Maximum Entropy Production (MEP). By using these connected variational principles we derived the optimal values of the relative angular position of the ATP-binding transition, which is in accord with the experimental best-fit angular position. MaxEnt and MEP predict an inverse relationship between the ATP synthase gearing ratio and the proton motive force in the region of approximate linearity of flux-force relationship far from equilibrium (the inflection point). For such proton-motive force molecular motor reacts with the highest speed to produce additional ATP or to save ATP molecules.

Relaxation process is a spontaneous transition of the isolated system between macroscopic states. It is assumed that the entropy increase associated with this process is the functional of dynamic variables (fluxes). This fact makes entropy production dynamical variable. It is shown within Jaynes' MaxEnt formalism that almost all possible microscopic fluxes are accompanied by maximum entropy production (MEP). Using Einsten's formula for probability of fluctation we obtain that the probability of the change of entropy is proportional to exponential function of the entropy change divided by the Boltzmann constant. The result derived for relaxation processes is extended to general non-equilibrium processes. This result is applied to th system close to equilibrium and the well known relationships between thermodynamic forces and fluxes are reproduced. The MEP principle applied to toy model of diffusion shows that standard diffusion law is an approximate one.

We start this talk posing the question, is there any physical principle that can serve as a selection principle in biology too? One of the first undertakings in this direction, conducted by Prigogine and Wiame [1] noticed correctly that biological processes are irreversible and as such should be described within irreversible thermodynamics. Since irreversible processes are characterized by entropy increase, they took the entropy production as the basic quantity for the description of biological processes and introduced the concept of dissipative structures. Prigogine suggested the principle of minimum entropy production (MinEP) [2] as a relevant principle for the biology, but the MinEP theorem describes the static head steady state, close to equilibrium state, when output flux and free-energy-transduction efficiency both vanish, an unrealistic and untenable situation for complex hierarchy of energy transductions in biological systems. Another principle is maximum entropy production (MaxEP) principle, which assumes that biological processes are accompanied by maximum possible entropy production. It looks like the antipode of the minimum entropy production principle. However, as it was stressed by Martyushev and Seleznev [3], these two principles should not be considered as contradictory or opposed to each other, since they include different constraints and different variable parameters. We argue here that MaxEP principle is more suitable than MinEP principle for a description of energy transduction in biological systems. There are two different approaches to MaxEP. Some authors assume that MaxEP can be derived from more fundamental principle. The most prominent author within this group is R. Dewar who made several attempts [4-6] to derive MaxEP from Jaynes' principle of maximum information entropy (MaxEnt). Another group assumes that the MaxEP principle is the fundamental one, which does not require some underlying principle as its foundation [3]. E. T. Jaynes, starting form Shannon information theory, launched the MaxEnt principle as universal principle which predicts the best unbiased distribution of [7, 8]. Our basic assumption is that the biological evolution of basic biochemical processes, like photosynthesis [10], free-energy transduction of ATP synthase [11] and catalytic cycle fluxes among enzyme internal functional states [12, 13] is accompanied by an entropy production increase and with the Shannon’s information entropy increase of enzyme states. In such a way the increase of entropy production is understood as being tightly coupled to biological evolution. We do not offer the MaxEP principle as an alternative to biological selection and evolution. Rather, it is the unique physical selection principle [9] which can be regarded as a good candidate for acting in concert with biological selection and evolution. [1] Prigogine, I., Wiame, J.M.: Biologie et thermodynamique des phénomènes irréversibles. Experientia 2, 451-453 (1946) [2] Prigogine, I.: Introduction to Thermodynamics of irreversible processes, John Wiley & Sons, New York (1967) [3] Martyushev, L.M., Seleznev, V.D.: Maximum entropy production principle in physics, chemistry and biology, Phys. Rep. 426, 1–45 (2006) [4] Dewar, R.C.: Information theory explanation of the fluctuation theorem, maximum entropy production and self- organized criticality in non-equilibrium stationary states. J. Phys. A: Math. Gen. 36, 631- 641 (2003) [5] Dewar, R.C.: Maximum entropy production and the fluctuation theorem. J. Phys. A: Math. Gen. 38, L371-L381 (2005) [6] Dewar, R.C., Maritan, A.: Beyond the Second Law: Entropy Production and Non-Equilibrium Systems, Eds. R. Dewar, R. Niven and K. Regenauer, Springer Verlag, to be published. [7] Jaynes, E.T.: Information theory and statistical mechanics. Phys. Rev. 106, 620–630 (1957) [8] Jaynes, E.T.: Information theory and statistical mechanics II. Phys. Rev. 108, 171–190 (1957) [9] Dewar, R.C.: Maximum entropy production and plant optimization theories. Phil. Trans. R. Soc. B 365, 1429-1435 (2010) [10] Juretić, D., Županović, P..:Photosynthetic models with maximum entropy production in irreversible charge transfer steps, J. Comp. Biol. Chem. 27, 541-553 (2003) [11] Dewar, R.C., Juretić, D., Županović, P.: The functional design of the rotary enzyme ATP synthase is consistent with maximum entropy production, Chem. Phys. Lett. 430, 177-182 (2006) [12] Dobovišek, A., Županović, P., Brumen, M., Bonačić-Lošić, Ž., Kuić, D., Juretić, D.: Enzyme kinetics and the maximum entropy production principle. Biophys. Chem. 154, 49-55 (2011) [13] Dobovišek, A., Županović, P., Brumen, M., Juretić, D.: Beyond the Second Law: Entropy Production and Non-Equilibrium Systems, Eds. R. Dewar, R. Niven and K. Regenauer, Springer Verlag, to be published.

Within isothermal analysis of Stirling engine analytical expressions for a pressure as a function of crank angle and for a work are found. The efficiency is numerical calculated for a wide range of temperatures. In contrast to the efficiency of the Carnot cycle, which is monotonically increasing function of the ratio of temperatures of heat baths, the efficiency of the Stirling engine has its maximum. The value of this maximum, as well as the corresponding ratio of heat baths temperatures, depends on a molar heat capacity of the working substance.

Algoritam maksimalne entropije i teorija informacije relevantni su za vremensku evoluciju makroskopskih sustava promatranu kao problem nepotpune informacije. Primjenjujući algoritam maksimalne entropije [1], [2] na predikciju vremenske evolucije zatvorenih sustava s Hamiltonovom dinamikom detaljno smo razmotrili dva različita pristupa [3]. Oba pristupa su zasnovana na Liouvilleovoj jednadžbi za uvjetnu distribuciju vjerojatnosti, definiranu za skup mikrostanja koji je pridružen skupu putanja određenih Hamiltonovim jednadžbama. Uz korištenje Shannonovog [4] koncepta uvjetne informacijske entropije, algoritam maksimalne entropije daje predikciju vremenske evolucije sustava konzistentnu s Liouvilleovom jednadžbom za uvjetnu distribuciju vjerojatnosti. U prvom pristupu predikcija vremenske evolucije je u potpunosti konzistentna s opisanim mikroskopskim ograničenjem u faznom prostoru. U drugom pristupu razmatrali smo nepotpunost informacije o mikroskopskoj dinamici sustava, na način koji je konzistentan s Jaynesovom formulacijom prediktivne statističke mehanike [5] i konceptom makroskopske reproducibilnosti za vremenski ovisne procese [6]. Maksimizacija uvjetne informacijske entropije uz Liouvilleovu jednadžbu uvedenu kao makroskopsko ograničenje, odnosno prosjek po faznom prostoru, vodi do gubitka korelacije između početnih putanja i konačnih mikrostanja. Informacijska entropija je teorijska gornja granica za uvjetnu informacijsku entropiju koja se dosiže jedino uz potpuni gubitak korelacije. U drugom pristupu, maksimizacija uvjetne informacijske entropije ekvivalentna je opisanom gubitku korelacije i vodi do odgovarajućeg gubitka informacije o mogućim mikrostanjima sustava. U skladu s izvornom idejom Jaynesa [2] ireverzibilnost je posljedica postupnog gubitka informacije o stanju sustava, što se danas razmatra kao mogući temeljni princip iza drugog zakona termodinamike [7], [8]. [1] Jaynes, E.T.: Information Theory and Statistical Mechanics. Phys. Rev. 106, 620-630 (1957) [2] Jaynes, E.T.: Information Theory and Statistical Mechanics. II. Phys. Rev. 108, 171-190 (1957) [3] Kuić, D., Županović, P., Juretić D.: Macroscopic time evolution and MaxEnt inference for closed systems with Hamiltonian dynamics. Submitted to Found. Phys. [4] Shannon C.E., Weaver, W.: The Mathematical Theory of Communication. University of Illinois Press, Urbana (1949) [5] Jaynes, E.T.: The Minimum Entropy Production Principle. Ann. Rev. Phys. Chem. 31, 579-601 (1980) [6] Jaynes, E.T.: Macroscopic Prediction. In: Haken, H. (ed.) Complex Systems - Operational Approaches in Neurobiology, Physics, and Computers, pp. 254-269. Springer-Verlag, Berlin (1985) [7] Duncan, T.L., Semura, J.S.: The Deep Physics Behind the Second Law: Information and Energy As Independent Forms of Bookkeeping. Entropy 6, 21-29 (2004) [8] Duncan, T.L., Semura, J.S.: Information Loss as a Foundational Principle for the Second Law of Thermodynamics. Found. Phys. 37, 1767-1773 (2007)

Possibility of application of MaxEnt algorithm to Hamiltonian dynamics of classical macroscopic closed systems is considered here in two approaches. The first approach is based on Liouville equations as strict microscopic constraints on time evolution of probability distributions in phase space. It is argued that this approach is equivalent to complete and objective information on microscopic dynamics. It is shown that it is possible to apply the MaxEnt algorithm consistently with this strict microscopic constraint, if the concepts of conditional probability distribution and respective conditional information entropy introduced by Shannon are utilized. Importance of phase space paths (solutions of Hamilton’s equations) is also emphasized. The second approach is also based on these concepts. Distinction between two approaches is in regarding the Liouville equation for conditional probability distribution as a macroscopic constraint in second approach, in accordance with Jaynes formulation of predictive statistical mechanics. It is shown that maximization of the conditional information entropy subject to this macroscopic constraint leads to a complete loss of correlation between the initial phase space paths and final microstates. During time evolution of the closed system, information entropy of the microstate probability distribution also becomes maximal. This happens indirectly, as a consequence of application of MaxEnt to the conditional information entropy. The former information entropy constitutes the upper bound on the latter, and the upper bound is attained only in case of the loss of correlation, or independence. It is shown therefore that if macroscopic time evolution is considered in this way these properties become equivalent.

A thermodynamic state of solvent and solution separated with an elastic semipermeable membrane, in the box with a fixed volume, is considered. It is shown that the minimum value of free energy is accompanied by the compression of the solution and tension of the solvent [1-4]caused by the transfer of solvent molecules into compartment with solution . The tensile state of the solvent is described in terms of negative pressure. Due to the elastic properties of membrane the difference of pressure between compartments is less than osmotic pressure calculated with van't Hoff equation. However, it is still enough high to uptake water in tall trees. It is proposed that osmosis at constant volume could be responsible for the water uptake in tall trees. Using plausible arguments it is shown that embolism is followed with refilling in this process [5-7]. In other words embolism repair is possible in osmotic process under constant volume. References [1] Hirsch, H.R. “Gedanken Experiment”: Negative Solvent Pressure in Osmosis. J. Biol. Physics, 1980, 8, 11-13. [2] Freeman , R.D. Comment on Hirsch’s “Gedanken Experiment”: Negative Solvent Pressure in Osmosis. T J. Biol. Physics, 1981, 9, 47. [3] Hirsch, H.R. Further comments on Negative Solvent Pressure in Osmosis. J. Biol. Physics, 1983, 11, p. 26. [4] Scholander, P.F. The Role of Solvent Pressure in Osmotic Systems. Botany, 1966, 55, 1407-1414. [5] Zimmermann, U., Schneider, H., Wegner, L. H. and Haase, A. Water ascent in tall trees: does evolution of land plants rely on a highly metastable state? New Phytol., 2004, 162, 575-615. [6] Holbrook, N.M. and Zwieniecki, M.A. Embolism Repair and Xylem Tension: Do We Need a Miracle? 1999, Plant Physiology, 1999, 120, 7–10, [7 ] Zwieniecki, M.A. and Holbrook, N.M. Confronting Maxwell’s demon: biophysics of xylem embolism repair. Trends in Plant Science, 2009, 14, 530-534.

A theoretical study of optimization of generalized reversible three state kinetic scheme of enzymatic reaction with respect to the Maximum Entropy Production Principle (MEPP) is presented. We calculated entropy production for each reaction transition of the kinetic scheme as a function either of forward or backward rate constant. We found that for each transition a unique maximum in entropy production exists, and, therefore, the optimal value of the corresponding rate constant could be calculated. Following idea that only second order rate constants are not subjected to evolution1, optimal values of rate constants predicted by MEPP principle are compared to known experimentally determined values for three different types of the enzyme beta-lactamase: beta- lactamase I, PC1 beta-lactamase and RTEM beta-lactamase2. Our results show that MEPP could be an appropriate principle for evolutionary optimization of enzymatic reactions. [1] Heinrich, R., Schuster, S., Holzhutter, H.-G.Mathematical analysis of enzyme reaction systems using optimization principles (1991) European Journal of Biochemistry, 201, 1-21. [2] H. Christensen, M.T. Martin and G. Waley β-lactamases as fully efficient enzymes. Determination of all the rate constants in the acyl-enzyme mechanism, (1990) Biochem. J. 266, 853-861.

The solvent in osmosis at constant volume is under negative pressure [1-4]. Negative pressure in solvent and osmotic pressure in solution are found as the function of the concentration of solute, geometry and elastic properties of the semipermeable membrane. It is known that the tensile state of the isolated solvent is not stable and cavitations turn it into stable state of coexisting gas and liquid. Using plausible arguments it is shown that this transition of the tensile solvent to coexisting phase can be reversible due to the elasticity of the semipermeable membrane. The reverse transition is diffusion and it is much slower that spontaneous direct transition. This mechanism is proposed as possible explanation of the embolism repair and refilling [5-7] in xylem-phloem system of land plants. References [1] Hirsch, H.R. “Gedanken Experiment”: Negative Solvent Pressure in Osmosis. J. Biol. Physics, 1980, 8, 11-13. [2] Freeman , R.D. Comment on Hirsch’s “Gedanken Experiment”: Negative Solvent Pressure in Osmosis. T J. Biol. Physics, 1981, 9, 47. [3] Hirsch, H.R. Further comments on Negative Solvent Pressure in Osmosis. J. Biol. Physics, 1983, 11, p. 26. [4] Scholander, P.F. The Role of Solvent Pressure in Osmotic Systems. Botany, 1966, 55, 1407-1414. [5] Zimmermann, U., Schneider, H., Wegner, L. H. and Haase, A. Water ascent in tall trees: does evolution of land plants rely on a highly metastable state? New Phytol., 2004, 162, 575-615. [6] Holbrook, N.M. and Zwieniecki, M.A. Embolism Repair and Xylem Tension: Do We Need a Miracle? 1999, Plant Physiology, 1999, 120, 7–10, [7 ] Zwieniecki, M.A. and Holbrook, N.M. Confronting Maxwell’s demon: biophysics of xylem embolism repair. Trends in Plant Science, 2009, 14, 530-534.

Mjerenja kutno razlučivih fotoemisijskih spektara (ARPES) kvazijednodimenzionalnog organskog vodiča TTF-TCNQ u normalnoj fazi pokazuju izostanak kvazičestičnih pobuđenja na kemijskom potencijalu i pojavu širokih struktura s jednodimenzionalnom disperzijom na većim energijama. Također, optička mjerenja su otkrila pojavu niskofrekvencijskog moda ($\approx10meV$) zajedno s modom frekvencije reda veličine veće ($\approx0.75eV$), koji su teorijski objašnjeni koristeći dielektrični formalizam prikladan za sisteme s više vrpci u aproksimaciji čvrste veze uzimajući dvije elektronske vrpce po lancima donorskih i akceptorskih molekula. Unutar ovog pristupa unutarvrpčana i međuvrpčana dipolna pobuđenja vezana monopolno-dipolnom interakcijom rezultiraju tim kolektivnim modovima. Krećući od zasjenjene dugodosežne Coulombove interakcije u gornjem pristupu koja sadrži dobivene kolektivne modove, koristimo $G_{;0};W_{;0};$ aproksimaciju za proračun jednočestične spektralne funkcije za vodljive jednodimenzionalne elektronske vrpce na akceptorskim i donorskim lancima. Pokazujemo da ovaj pristup objašnjava spektar TTF-TCNQ-a dobiven kutno razlučivom fotoemisijom. Konačno, u ovom radu proširujemo izneseni pristup objašnjenju optičkih svojstava TTF-TCNQ-a na kvazidvodimenzionalni vodič $\beta''-(BEDT-TTF)_2SF_5CH_2CF_2SO_3$ čija optička mjerenja također pokazuju niskoenergijski kolektivni mod.

Xylem is modelled as the closed system coupled to the root and phloem via semi-permeable membranes. The importance of the fixed volume of the xylem in the osmotic process between xylem and phloem is analysed. It is shown that negative pressure, a concept accepted as a driving force for the upward xylem water transport in many scientific discussions, is closely related to this process. It is shown that there is a self-consistency between decrease of the pressure in the xylem and increase of the concentration of the sugar in phloem, respectively, in dependence of the height. [1] Zimmermann, U., H. Schneider, L. H. Wegner, and A. Haase. 2004. Water ascent in tall trees: does evolution of land plants rely on a highly metastable state? New Phytol. 162:575-615. [2] Lampinen, M. J., and T. Noponen. 2003. Thermodynamic analysis of the interaction of the xylem water and phloem sugar solution and its significance for the cohesion theory. J. Theor. Biol. 224:285-298.

The resonance of the Wilberforce pendulum is defined as the state of the maximum period of beats. The plausible assertion that resonance is characterized by the equal values of both the frequencies of longitudinal and torsion vibrations is proven. Although a coupling constant between longitudinal and torsion vibrations determines the frequencies of normal modes it plays no role in the definition of the resonance state.

MEPP is a physical principle, widely used for quantitative explanation of non equilibrium phenomena in physics, chemistry and biology [1-3]. Here, we applied MEPP to study two and three state reversible Michaelis-Menten kinetic schemes of enzymatic reactions. By applying constraints as a diffusional limit for kinetic constants, constant free energy differences between enzymatic states and constant thermodynamic force, we calculated Shannon information entropy and entropy production of the entire reaction system as a function of forward rate constants. We found maxima in the net steady-state metabolic flux, total entropy production and Shannon entropy for equal values of forward rate constants. Moreover, for these values an analytical expression derived gives a relation between substrate and product concentrations at which enzymes operate in the optimal way. In conclusion, we demonstrated that MEPP is an appropriate selection principle for evolutionary optimization of enzymes. [1]Martyushev, L.M., Seleznev V.D. (2006) Physics Reports 426: 1-45. [2]Juretic, D., Zupanovic, P. (2003) CBC 27: 541 - 553. [3]Dewar, R.C., Juretic, D., Zupanovic, P. (2006) Chem. Phys. Lett. 430: 177-182.

Theoretical studies from past years show that the principle of maximum entropy production (MEP) is widely accepted optimization principle used for quantitative explanation of many non-equilibrium phenomena [1]. From this point of view, the MEP principle is also of basic importance for understanding of biological systems, which operate in non-linear regime, far away from equilibrium state. For example, it was shown [2] that a rotary enzyme ATP-synthase produces entropy at a maximal value. The MEP principle was applied to analysis of one of the most important transition in the entire reaction scheme of this enzyme and it was able to predict the values of kinetic parameters associated with this transition, that are close to experimental values. We prove analytically in this work that maximal entropy production and optimal forward reaction rate constant can be associated with any chosen transition connecting two given basic free energy levels [3] in the case of an arbitrarily complex steady-state reaction scheme of an enzyme that can exist in a finite number of discrete states. The MEP principle is then applied to the analysis of the three-state kinetic scheme of enzymatic reaction considered in terms of generalized Michaelis-Menten steady-state kinetics. These three states are governed by three backward and three forward kinetic constants. By applying as constraints the maximal value of the product of forward reaction rate constants, constant free energy differences between enzymatic states, and constant thermodynamic force, we calculated Shannon entropy and entropy production as a function of any chosen couple of two forward rate constants. We show numerically and analytically that under these conditions there is a maximum in the net substrate to product steady-state flux, total entropy production and Shannon entropy of the system for the same optimal value of all three forward reaction rate constants. In conclusion, this is the first theoretical study were maximum entropy production is found for any given transition between two functionally important enzyme states and for the entire reaction scheme of an enzymatic reaction plus surrounding bath with substrates and products. We point out that the MEP principle and maximum Shannon entropy principle [4] can be treated as powerful physical selection principles for the evolutionary optimization of enzymes. [1] Martyushev, L.M., Seleznev V.D. Maximum entropy production principle in physics, chemistry and biology. (2006) Physics Reports 426: 1-45. [2] Dewar, R.C., Juretić, D., Županović, P. The functional design of the rotary enzyme ATP synthase is consistent with maximum entropy production.(2006) Chem. Phys. Lett. 430: 177-182. [3] Hill, T.L. Free Energy Transduction in Biology, Academic Press, London, 1977 [4] Jaynes, E.T. Probability Theory, Cambridge Univ. Press, 2003.

Theoretical studies from past years show that the principle of maximum entropy production (MEP) is widely accepted optimization principle used for quantitative explanation of many non-equilibrium phenomena [1]. From this point of view, the MEP principle is also of basic importance for understanding of biological systems, which operate in non-linear regime, far away from equilibrium state. For example, it was shown [2] that a rotary enzyme ATP-synthase produces entropy at a maximal value. The MEP principle was applied to analysis of one of the most important transition in the entire reaction scheme of this enzyme and it was able to predict the values of kinetic parameters associated with this transition, that are close to experimental values. In the present contribution the MEP principle is applied to general analyses of the three-state kinetic scheme of enzymatic reaction considered in terms of Michaelis-Menten kinetics. These three states are governed by three backward and three forward kinetic constants. The system was elaborated in terms of the Hills graph theory [3]. By applying additional constraints for maximal values of forward reaction rate constants as well as constant free energy differences between the enzyme states, we calculated Shannon informational entropy and entropy production as a function of two forward rate constants of the second and third state. We can show numerically that under these conditions there is a maximum in the total entropy production of the system and Shannon informational entropy of the system. In conclusion, this is the first theoretical study were the maximum of total entropy production is found for the entire reaction scheme of enzymatic reaction. We point out that the MEP principle can be treated as a powerful physical selection principle for the evolutionary optimization of enzymes. References [1] Martyushev, L.M., Seleznev V.D. Maximum entropy production principle in physics, chemistry and biology. 2006 Physics reports 426: 1-45. [2] Dewar, R.C., Juretić, D., Županović, P. The functional design of the rotary enzyme ATP synthase is consistent with maximum entropy production.2006. CPL 430: 177-182. [3] T.L. Hill. Free energy transduction in biology, Academic Press, London, 1

Pri izduženju cilindrična opruga se uvija i smanjuje svoj promjer. Na taj način sprežu se uzdužna i torzijska titranja opruge (1). Wilberforceov oscilator je uređaj pomoću kojega se demonstrira ova sprega. U slučaju kada si frekvencije uzdužnog i torzijskog načina titranja jednake javlja se rezonancija. Mehanička energija uzdužnog titranja prijelazi potpuno u mehaničku energiju torzijskog titranja i obratno (1-5). Izložene su fenomenološke osnove rada ovog oscilatora. Wilberforceov oscilator je izuzetno efektan pokus za prikaz rezonancije. Demonstrirat će se rad ovog uređaja. [1] Köpf, U., Wilberforce’s pendulum revisited, Am. J. Phys. 58, 833-837 (1990). [2] Wilberforce, L. R., On Vibrations of Loaded Spiral Spring, Phil. Mag. 38, 386-392 (1894). [3] Love, A. E. H., A Treatise on the Mathematical Theory of Elasticity, (Dover Publications, New York, 1944). [4] Sommerfeld, A., Mechanics of Deformable Bodies, (AP, New York, 1950). [5] Berg, R. E. and Marshall, T. S., Wilberforce pendulum oscillations and normal modes, Am. J. Phys. 59, 32-39 (1991).

The thermodynamic point of view is used to consider bacterial chemotaxis in a capillary tube, and entropy production is calculated for a migrating band of bacteria. It is found that the speed of the migrating band is a decreasing function of the starting concentration of the metabolizable attractant. The experimentally found dependence of speed on the starting concentration of galactose, glucose and oxygen is fitted with power law functions. It is found that the corresponding exponents lie within the theoretically predicted interval. The effect of the reproduction of bacteria on band speed is considered, too. The acceleration of the band, owing to the reproduction rate of bacteria, is predicted. The relationship between chemotaxis, the maximum entropy production principle and the formation of self-organizing structure is briefly discussed.

Time development of the isolated ring of linearly coupled oscillators is found. The initial states are elements of the subensemble that is characterized with the fixed, nonuniform, distribution of oscillators amplitudes and equally weighted phases. Time development of the system, averaged over phase space, is calculated. The energies of oscillators are quantized. The probability of finding quantum of energy at a certain oscillator is proportional to the energy of an oscillator. Entropy of the system is described in terms of the distribution of energy quanta using the Shannon’ s definition of the information [1]. Numerical calculations show that entropy has a tendency to increase in accordance with the second law of thermodynamics. No entropy fluctuations during the relaxation process have been found. Entropy increases smoothly until the system reaches equilibrium. Then appear fluctuations. Single microscopic state is considered. Following Zurek [2] it is assumed that entropy can be assigned to this state. It is found that the time development of entropy is qualitatively same as in the case of the subensemble, i.e. entropy shows no fluctuation during the relaxation process. The distribution of the energies is calculated in the equilibrium state and it is found that it obeys Boltzmann distribution. [1] C. E. Shannon, The Mathematical theory of communication, University of Illinois Press, Chicago, 1998. [2] W. H. Zurek, Phys. Rev. A 40, 4731 (1989). [3] E. T. Jaynes, Phys. Rev. 106, 620 (1957). [4] E. T. Jaynes, Phys. Rev. 108, 171 (1957).

The thermodynamic point of view is used to consider bacterial chemotaxis in a capillary tube, and entropy production is calculated for a migrating band of bacteria. It is found that the speed of the migrating band is a decreasing function of the starting concentration of the metabolizable attractant. The experimentally found dependence of speed on the starting concentration of galactose, glucose and oxygen is fitted with power law functions. It is found that the corresponding exponents lie within the theoretically predicted interval. The effect of the reproduction of bacteria on band speed is considered, too. The acceleration of the band, owing to the reproduction rate of bacteria, is predicted. The relationship between chemotaxis, the maximum entropy production principle and the formation of self-organizing structure is briefly discussed.

The question connecting physics and biology, which has never been resolved in the literature, is about interdependence of biological and thermodynamic evolution: are biological macromolecules and living cells optimized to produce as little or as much entropy as possible. Related question is about Shannon’ s (information) entropy: are present-day enzymes designed by evolution to have minimal or maximal information entropy. These questions will be discussed for steady state enzyme kinetics applied in modeling bacterial photosynthesis, the molecular motor of ATP synthase and generalized Michaelis-Menten mechanism for enzymatic reactions. We shall examine the hypothesis that natural evolution in biology can be predicted as well as evolution in thermodynamics by using the same rules of statistical mechanics, when assumption is made that the most probable behavior in each case is selected by the principle of maximum Shannon entropy and one of its corollaries – Maximum Entropy Production Principle.

Lanac slabo vezanih oscilatora je točno rješiv problem. U skladu s Jaynes-ovim pristupom [1, 2], po kome nije moguće odrediti početno mikroskopsko stanje sustava, promatrana je relaksacija sustava s početnim stanjem koje je usrednjeno preko ansambla faza. Svi elementi ovog ansambla imaju iste početne amplitude. Diskretizirani su kvadrati amplituda oscilatora. Jedinica kvantizacije se promatra kao nasumična veličina. U skladu s Zurekovim [3] pristupom definirana je entropija neravnotežnog stanja preko Shannonove definicije entropije [4] kao negativne informacije. Numerički je izračunata entropija kao funkcija vremena. Entropija se kao monotono rastuća funkcija vremena asmptotski približava maksimalnoj vrijednosti. Drugim rječima vremenski razvoj sustava slabo vezanih oscilatora u skladu je s drugim zakonom termodinamike. Reference: [1]E.T. Jaynes, Phys. Rev. 106, 620 (1957). [2]E.T. Jaynes, Phys. Rev. 108, 171 (1957). [3]W. H. Zurek, Phys. Rev. A 40, 4731 (1989). [4] C.E. Shannon, The Mathematical theory of communication (University of Illinois Press, Chicago, 1998)

Spontaneous transitions of an isolated system from one macroscopic state to another (relaxation processes) are accompanied by a change of entropy. Following Jaynes’ MaxEnt formalism it is shown that practically all possible microscopic developments of the system, within fixed time interval, are accompanied by the maximum possible entropy change. In other words relaxation processes are accompanied by maximum entropy production. Applying the maximum entropy production principle to the systems close to the equilibrium a complete picture of linear non-equilibrium thermodynamics is derived. A logarithmic dependence of the diffusion (heat) flux on the gradient of concentration (temperature) is predicted. For small gradients of concentration or temperature, the theory reduces to the standard laws of gas diffusion and thermal conductivity of insulators.

The evolution in physics leads from less to more probable system behaviour under given constraints, while evolution in biology is usually described as leading to more complex and less probable systems. In this work we examine the hypothesis that natural evolution in biology can be predicted as well as evolution in thermodynamics by using the same rules of statistical mechanics. We assume that the most probable behaviour in each case is selected by the principle of maximum Shannon entropy (MaxEnt) and one of its corollaries – Maximum Entropy Production (MEP). For a kinetic model of ATP synthase MaxEnt/MEP correctly predicts: a) the angular position of the catalytic dwell ; b) the gearing ratio (the H+/ATP stoichiometry) dependence on the proton-motive force ; c) approximate flux-force linearity far from equilibrium. The ability to predict the design features of bio-macromolecules evolved through natural selection (1, 2) points toward the MaxEnt/MEP as common predictive framework in physics and biology (3, 4).

Relaxation process is the spontaneous transition of an isolated system from one to another macroscopic state. It is assumed that the entropy increase associated with this process is a functional of dynamical variables (fluxes). This fact makes entropy production dynamical variable. It is shown within Jaynes’ MaxEnt formalism that almost all possible microscopic fluxes are accompanied by maximum entropy production. Using Einstein’ s formula for probability of fluctuation we obtain that the probability of the change in entropy is proportional to the exponential function of the entropy change divided by the Boltzmann constant. This result is applied to the system close to the equilibrium and the well-known relationships between thermodynamic forces and fluxes are reproduced.

The spectral properties of quasi-one dimensional metals at the higher energies of the order of plasmon and/or bandwidth energy are investigated by applying the standard co-called G0W0 approximation in the calculation of the dressed electron Green's function with the screened long-range three-dimensional Coulomb interaction obtained within the random phase approximation (RPA). First we assume a strictly one-dimensional electron band due to a finite transfer integral along the chains, and in the second step we extend the consideration to a more realistic case of electron band with a finite interchain hopping. As long as there is no interchain hopping the spectral function doesn't have low-energy quasi-particle, but contains broad structure with the width of the order of plasmon energy. Such spectral properties are in qualitative agreement with ARPES spectra of Bechgaard salts.

Maximum entropy production (MEP) principle has been derived recently in a ground-breaking work of Dewar (1), who used an information theoretical formulation of non-equilibrium statistical mechanics. The MEP principle states that the MEP state is reproducibly selected because it is the most probable non-equilibrium steady state compatible with given external constraints. In the linear region, for the case of an electric network, MEP is equivalent to the Kirchhoff’ s loop law when overall energy conservation is assumed (2). In the case of heat conduction in an anisotropic crystal, MEP is equivalent to the Onsager-Rayleigh principle of the “ least dissipation of energy” (3). In the nonlinear region, MEP provides generalization of Onsager’ s theory for situations and systems far from equilibrium. For instance, when modeling global climate MEP predicts that winds and currents driven by thermal gradients establish themselves as the most effective heat transport from the warmer tropics to the colder poles which maximizes the entropy production. We shall discuss what would be the best mode of MEP application in biochemistry (4, 5) and how realistic are selected states with maximum information entropy and maximum dissipation. Natural selection for the coexistence of ordered and dissipative regions far from equilibrium can be understood as an expression of the same basic concept, namely, selection of the most probable state, which produces and exports more entropy to the environment than a purely dissipative "soup". To conclude, recent research at the University of Split points out toward maximum entropy production principle as a guideline to modeling complex system, including predicting its evolution, irrespective of system scale, nature, or its distance to equilibrium. 1)Dewar, R. C. J. Phys. A: Math. Gen. 38, L371– L381 (2005). 2)Županović, P., Juretić, D., Botrić, S. Phys. Rev. E. 70, 0561108 (2004). 3)Županović, P., Juretić, D., Botrić, S. Fizika, in press (2005). 4)Županović, P., Juretić, D. Croat.Chem. Acta 77, 561-571 (2004). 5) Juretić, D., Županović, P. Comput.Biol.Chem. 27, 541-553.

Tijekom zadnje dvije godine Zavod za fiziku Fakulteta prirodoslovno-matematičkih znanosti u Splitu u velikoj mjeri je preobrazio i modernizirao nastavu fizike u skladu s Bolonjskom deklaracijom. Uz nastavničke dvopredmetne studije (Fizika i informatika te Matematika i fizika) sada nudimo i razna diplomska usmjerenja studijske grupe Fizika te studij inženjerske fizike. U Republici Hrvatskoj smjerovi Astrofizike, Biofizike, Fizike okoliša te Računarske fizika, koje je moguće izabrati u okviru Diplomskog studija fizike, kao i studij Inženjerske fizike (sa smjerovima Termodinamički uređaji i Mehanički Sustavi) nude se samo na Sveučilištu u Splitu. Kvalitetu tih studija osiguravaju nastavnici visoke međunarodne prepoznatljivosti i široka suradnja s znanstveno-nastavnim ustanovama u zemlji i inozemstvu. Opisali smo i konkretni primjer kako je modernizacija predmeta opće fizike (mehanika i eletromagnetizam) u prvoj godini studija dovela do povećanog interesa studenata za fiziku i do znatno većeg postotka prijelaza studenata u drugu godinu studija (porast od oko 30% na oko 70%).

In the biological evolution of enzymatic reaction rates, natural selection has modified various kinetic and structural parameters within the constraints set by the physics of enzymatic catalysis. Here we consider this selection process from a thermodynamic viewpoint. Using a previously calibrated kinetic model of ATP synthesis and hydrolysis, we show that a selection principle of maximum entropy production (MEP) allows accurate prediction of the angular position of the binding transition state of rotary F0F1-ATP synthase in spinach chloroplasts. The same angular position also maximises the Shannon entropy of the enzyme state population distribution. MEP and maximum enzyme state entropy also coincide with the optimal functioning of a general single-cycle Michaelian reaction scheme. Our results support the recent theoretical proposal that MEP is a statistical selection principle underlying a wide range of far-from-equilibrium processes across physics and biology.

We Used Jaynes’ MaxEnt formalism to consider spontaneous transition of isolated system between two macroscopic stastes. It is shown that vast majority of paths in phase space are characterized by maximum possible entropy change. This result is compared with the Einstein's formula for fluctation probability.

The standard derivation of Kirchhoff’ s voltage law is based on the assumption that the first law of thermodynamics (sum of products of applied EMF’ s and corresponding currents = heat dissipation) is satisfied within each loop. In contrast, we start from the fact that first law of thermodynamics applies globally to the electric circuit as a whole. It is then shown that Kirchhoff’ s voltage law may be derived as an extremum of the system entropy production constrained by the first law applied globally, and that the extremum is a maximum. Thus the stationary distribution of currents in a linear electric circuit is governed by the principle of maximum entropy production.

An application of the Onsager's principle of the least dissipation of energy to the problem of the heat conduction in an anisotropic crystal is revisited. In the case of a steady spatial distribution of the temperature it is shown that this principle is equivalent to the principle of maximum entropy production. This means that components of the actual heat flux are distributed in such a way as to achieve the state of the maximum entropy production., Obnavljamo pitanje primjene Onsagerovog principa najmanjeg rasipanja slobodne energije na problem vođenja topline u anizotropnom kristalu. U slučaju stalne prostorne raspodjele temperature, pokazuje se da je taj princip jednakovaljan principu najveće brzine rasta entropije. Stoga se komponente toplinskog toka raspodijele tako da se postigne stanje najveće brzine rasta entropije.

Transport Boltzmann euation is solved in the case of the stationary non-equilibrium processes close to the equilibrium. It is shown that among all possible fluxes, which ensure that the entropy produced in molecular collisions is equal to the entropy exported from the system, the actual fluxes are those that maximize the entropy production.