## Tesi etd-06262018-112543 |

Thesis type

Tesi di laurea magistrale

Author

FONTANA, PIERPAOLO

URN

etd-06262018-112543

Title

Scaling behaviour of Ising systems at first-order transitions

Struttura

FISICA

Corso di studi

FISICA

Commissione

**relatore**Prof. Vicari, Ettore

Parole chiave

- finite-size scaling
- fisica statistica
- transizioni di fase del primo ordine

Data inizio appello

19/07/2018;

Consultabilità

secretata d'ufficio

Riassunto analitico

The properties of finite-size systems near a phase transition are characterized by a finite-size scaling (FSS) behaviour. In the case of continuous phase transitions this behaviour is described by power laws with universal critical exponents, in the sense that they depend only by global features of the system (e.g. symmetries of the hamiltonian, number of spatial dimensions); the case of first-order phase transitions (FOTs) is different and in a way more complicated, since there is no systematic and well-established theory that describes them. The most studied case in literature is that of finite systems with periodic boundary conditions (PBC): here the finite-size effects are characterized by a power law behaviour with exponents related to the space dimension of the system. However, in general finite-size effects at FOTs depends on the geometry and on the boundary conditions considered.

Thus it is interesting to show how the scaling behaviour of a finite system is modified in correspondence of different boundary conditions, both for equilibrium and off-equilibrium properties.

In this work we investigate how the behaviour of the 2D Ising model, presenting a field-driven FOT, changes in correspondence of various boundary conditions by using Monte Carlo simulations and FSS analysis in the case of a purely relaxational dynamics.

In the case of PBC a dynamic FSS theory for the dynamic behaviour of coexisting phases is developed. The behaviour of the system in the coexistence region is characterized by a time scale T(L) that increases exponentially with L. Since these boundary conditions promote the formation of two planar interfaces that separate the two phases, the time scale is expected to be T(L)=(L^α)e^(σL), where σ=2βK, K is the planar interface tension and α is an appropriate exponent. In the thesis we extend these studies to open boundary conditions (OBC): in particular we show that the time scale is exponential as well as before, but σ is a half of the previous value, since OBC allow the formation of a single interface that separates the two coexisting phases. For everything else, the dynamic FSS theory developed for PBC can be applied also in this case. Data obtained with numerical simulations for a purely relaxational dynamics confirm our general theory.

As a second step, we consider the case of opposite fixed boundary conditions (OFBC), i.e. along the vertical direction there are PBC and along the horizontal one the spins on the boundary of the lattice interact with a column of fixed +1 on the right side and -1 on the left side (or vice versa). For these boundary conditions the equilibrium dynamic exponent z of the purely relaxational dynamic is evaluated: the dynamics is related to the interface enforced by MBC, which is not stationary and gives rise to a power-law, i.e. T(L) is proportional to L^z. This power-law behaviour is observed as well as an estimate of z is obtained with the help of numerical simulations, even though the latter result can not be compared with any theoretical value. It is also observed the scaling of the average magnetization as a function of r_1=hL^2, underlining the connection between the scaling function of the magnetization and the position of the interface within the lattice.

Finally equal fixed boundary conditions (EFBC) are considered (all the spins on the boundary are +1 or -1). In the case of a square lattice of linear dimension L and in the coexistence region the average magnetization scales with r_1=hL^ε, with ε≈1.7. Furthermore we have done the same analysis for a lattice with slablike geometry (a 2LxL lattice) presenting EFBC along the horizontal direction and PBC along the vertical one (and vice versa): in this case the magnetization in the coexistence region scales with r_1=hL^2.

These scaling behaviours may be related to the nature of the domain walls separating the coexistent phases of the system: in the case of the square lattice with FBC the interface presents a curvature, while in the other case boundary conditions promote the formation of a planar interface (without curvature).

The effects discussed in the thesis should be observable in various physical situations, where a FOT is approached by varying the external parameters of a small finite system: as a case of current physical interest we mention the experimental search of FOTs of the quark-gluon plasma in heavy-ion collision experiments.

All these cases highlight the fact that finite systems presenting FOTs are particularly sensitive to the change of boundary conditions, both for equilibrium and dynamic properties: the understanding of these finite-size effects is important to interpret in the right way experiments and results obtained with numerical simulations of finite-size systems close to the transition point.

In the view of potential future developments, it could be interesting also to study the off-equilibrium behaviour of Ising systems at FOTs: one may consider an off-equilibrium dynamics driven by a time-dependent magnetic field, in order to see if a nontrivial scaling behaviour is observed when the transition point is slowly crossed.

Thus it is interesting to show how the scaling behaviour of a finite system is modified in correspondence of different boundary conditions, both for equilibrium and off-equilibrium properties.

In this work we investigate how the behaviour of the 2D Ising model, presenting a field-driven FOT, changes in correspondence of various boundary conditions by using Monte Carlo simulations and FSS analysis in the case of a purely relaxational dynamics.

In the case of PBC a dynamic FSS theory for the dynamic behaviour of coexisting phases is developed. The behaviour of the system in the coexistence region is characterized by a time scale T(L) that increases exponentially with L. Since these boundary conditions promote the formation of two planar interfaces that separate the two phases, the time scale is expected to be T(L)=(L^α)e^(σL), where σ=2βK, K is the planar interface tension and α is an appropriate exponent. In the thesis we extend these studies to open boundary conditions (OBC): in particular we show that the time scale is exponential as well as before, but σ is a half of the previous value, since OBC allow the formation of a single interface that separates the two coexisting phases. For everything else, the dynamic FSS theory developed for PBC can be applied also in this case. Data obtained with numerical simulations for a purely relaxational dynamics confirm our general theory.

As a second step, we consider the case of opposite fixed boundary conditions (OFBC), i.e. along the vertical direction there are PBC and along the horizontal one the spins on the boundary of the lattice interact with a column of fixed +1 on the right side and -1 on the left side (or vice versa). For these boundary conditions the equilibrium dynamic exponent z of the purely relaxational dynamic is evaluated: the dynamics is related to the interface enforced by MBC, which is not stationary and gives rise to a power-law, i.e. T(L) is proportional to L^z. This power-law behaviour is observed as well as an estimate of z is obtained with the help of numerical simulations, even though the latter result can not be compared with any theoretical value. It is also observed the scaling of the average magnetization as a function of r_1=hL^2, underlining the connection between the scaling function of the magnetization and the position of the interface within the lattice.

Finally equal fixed boundary conditions (EFBC) are considered (all the spins on the boundary are +1 or -1). In the case of a square lattice of linear dimension L and in the coexistence region the average magnetization scales with r_1=hL^ε, with ε≈1.7. Furthermore we have done the same analysis for a lattice with slablike geometry (a 2LxL lattice) presenting EFBC along the horizontal direction and PBC along the vertical one (and vice versa): in this case the magnetization in the coexistence region scales with r_1=hL^2.

These scaling behaviours may be related to the nature of the domain walls separating the coexistent phases of the system: in the case of the square lattice with FBC the interface presents a curvature, while in the other case boundary conditions promote the formation of a planar interface (without curvature).

The effects discussed in the thesis should be observable in various physical situations, where a FOT is approached by varying the external parameters of a small finite system: as a case of current physical interest we mention the experimental search of FOTs of the quark-gluon plasma in heavy-ion collision experiments.

All these cases highlight the fact that finite systems presenting FOTs are particularly sensitive to the change of boundary conditions, both for equilibrium and dynamic properties: the understanding of these finite-size effects is important to interpret in the right way experiments and results obtained with numerical simulations of finite-size systems close to the transition point.

In the view of potential future developments, it could be interesting also to study the off-equilibrium behaviour of Ising systems at FOTs: one may consider an off-equilibrium dynamics driven by a time-dependent magnetic field, in order to see if a nontrivial scaling behaviour is observed when the transition point is slowly crossed.

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