Self expanding lattices

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### Self expanding lattices

odeint supports changes of the state size during integration if a state_type is used which can be resized, like `std::vector`. The adjustment of the state's size has to be done from outside and the stepper has to be instantiated with `always_resizer` as the template argument for the `resizer_type`. In this configuration, the stepper checks for changes in the state size and adjust it's internal storage accordingly.

We show this for a Hamiltonian system of nonlinear, disordered oscillators with nonlinear nearest neighbor coupling.

The system function is implemented in terms of a class that also provides functions for calculating the energy. Note, that this class stores the random potential internally which is not resized, but rather a start index is kept which should be changed whenever the states' size change.

    typedef vector< double > coord_type;
    typedef pair< coord_type , coord_type > state_type;

    struct compacton_lattice
    {
        const int m_max_N;
        const double m_beta;
        int m_pot_start_index;
        vector< double > m_pot;

        compacton_lattice( int max_N , double beta , int pot_start_index )
            : m_max_N( max_N ) , m_beta( beta ) , m_pot_start_index( pot_start_index ) , m_pot( max_N )
        {
            srand( time( NULL ) );
            // fill random potential with iid values from [0,1]
            boost::mt19937 rng;
            boost::uniform_real<> unif( 0.0 , 1.0 );
            boost::variate_generator< boost::mt19937&, boost::uniform_real<> > gen( rng , unif );
            generate( m_pot.begin() , m_pot.end() , gen );
        }

        void operator()( const coord_type &q , coord_type &dpdt )
        {
            // calculate dpdt = -dH/dq of this hamiltonian system
            // dp_i/dt = - V_i * q_i^3 - beta*(q_i - q_{i-1})^3 + beta*(q_{i+1} - q_i)^3
            const int N = q.size();
            double diff = q[0] - q[N-1];
            for( int i=0 ; i<N ; ++i )
            {
                dpdt[i] = - m_pot[m_pot_start_index+i] * q[i]*q[i]*q[i] -
                        m_beta * diff*diff*diff;
                diff = q[(i+1) % N] - q[i];
                dpdt[i] += m_beta * diff*diff*diff;
            }
        }

        void energy_distribution( const coord_type &q , const coord_type &p , coord_type &energies )
        {
            // computes the energy per lattice site normalized by total energy
            const size_t N = q.size();
            double en = 0.0;
            for( size_t i=0 ; i<N ; i++ )
            {
                const double diff = q[(i+1) % N] - q[i];
                energies[i] = p[i]*p[i]/2.0
                    + m_pot[m_pot_start_index+i]*q[i]*q[i]*q[i]*q[i]/4.0
                    + m_beta/4.0 * diff*diff*diff*diff;
                en += energies[i];
            }
            en = 1.0/en;
            for( size_t i=0 ; i<N ; i++ )
            {
                energies[i] *= en;
            }
        }

        double energy( const coord_type &q , const coord_type &p )
        {
            // calculates the total energy of the excitation
            const size_t N = q.size();
            double en = 0.0;
            for( size_t i=0 ; i<N ; i++ )
            {
                const double diff = q[(i+1) % N] - q[i];
                en += p[i]*p[i]/2.0
                    + m_pot[m_pot_start_index+i]*q[i]*q[i]*q[i]*q[i] / 4.0
                    + m_beta/4.0 * diff*diff*diff*diff;
            }
            return en;
        }

        void change_pot_start( const int delta )
        {
            m_pot_start_index += delta;
        }
    };

The total size we allow is 1024 and we start with an initial state size of 60.

    //start with 60 sites
    const int N_start = 60;
    coord_type q( N_start , 0.0 );
    q.reserve( max_N );
    coord_type p( N_start , 0.0 );
    p.reserve( max_N );
    // start with uniform momentum distribution over 20 sites
    fill( p.begin()+20 , p.end()-20 , 1.0/sqrt(20.0) );

    coord_type distr( N_start , 0.0 );
    distr.reserve( max_N );

    // create the system
    compacton_lattice lattice( max_N , beta , (max_N-N_start)/2 );

    //create the stepper, note that we use an always_resizer because state size might change during steps
    typedef symplectic_rkn_sb3a_mclachlan< coord_type , coord_type , double , coord_type , coord_type , double ,
            range_algebra , default_operations , always_resizer > hamiltonian_stepper;
    hamiltonian_stepper stepper;
    hamiltonian_stepper::state_type state = make_pair( q , p );

The lattice gets resized whenever the energy distribution comes close to the borders `distr[10] > 1E-150`, `distr[distr.size()-10] > 1E-150`. If we increase to the left, `q` and `p` have to be rotated because their resize function always appends at the end. Additionally, the start index of the potential changes in this case.

    double t = 0.0;
    const double dt = 0.1;
    const int steps = 10000;
    for( int step = 0 ; step < steps ; ++step )
    {
        stepper.do_step( boost::ref(lattice) , state , t , dt );
        lattice.energy_distribution( state.first , state.second , distr );
        if( distr[10] > 1E-150 )
        {
            do_resize( state.first , state.second , distr , state.first.size()+20 );
            rotate( state.first.begin() , state.first.end()-20 , state.first.end() );
            rotate( state.second.begin() , state.second.end()-20 , state.second.end() );
            lattice.change_pot_start( -20 );
            cout << t << ": resized left to " << distr.size() << ", energy = " << lattice.energy( state.first , state.second ) << endl;
        }
        if( distr[distr.size()-10] > 1E-150 )
        {
            do_resize( state.first , state.second , distr , state.first.size()+20 );
            cout << t << ": resized right to " << distr.size() << ", energy = " << lattice.energy( state.first , state.second ) << endl;
        }
        t += dt;
    }

The `do_resize` function simply calls `vector.resize` of `q` , `p` and `distr`.

    void do_resize( coord_type &q , coord_type &p , coord_type &distr , const int N )
    {
        q.resize( N );
        p.resize( N );
        distr.resize( N );
    }

The full example can be found in resizing_lattice.cpp